Why a Buddhist prays at Hindu temple which is situated in another country and prays for the betterment of his own country?

Firstly, as ChrisW mentioned, a person might do this to please others. For example, in this case, a Buddhist politician may partake in a ceremony in another religion's place of worship to make other people happy. For example, here is a news article on Indian PM Narendra Modi visiting a Hindu temple, a Buddhist temple and a mosque in Singapore. I'm not sure if he participated in any religious ceremonies.

Secondly, a person might do this to fit in. For example, a member of Buddhism.SE who is a practising lay Buddhist from a Muslim background, wrote an answer here (which I think I won't provide the link for), stating that he did not openly declare himself to be a Buddhist and instead lives an outward life appearing to be a Muslim, whenever he meets his relatives. I'm only paraphrasing the answer, according to my understanding of it.

Thirdly, lay Buddhists and even some monks may participate in what appears to be superstitious rituals, even if the Buddha discouraged this. Just because some people identify themselves as Buddhist, doesn't mean that they are aware of the Dhamma (teachings).

For example, from this article:

This leads to the highlight of the day - chanting and blessings by monks and ending the short session (which lasts no more than 5 minutes) with a shower of "holy water".

The monks softly chants a Thai variety of the Ti-Ratana Vandana (homage to the Triple Gem), five precepts and punnanumodana (sharing of merits). For most however, these chants are exotic and spiritual, and are perceived to have "supra mundane" powers. It is no wonder then that many devotees place their personal belongings - amulets, religious icons and even car keys - onto a tray held by the monk who performs the chanting.

Their belief - mistaken or otherwise - is that these chants contain "spiritual powers" which are then transcended into their items, which when worn or used, protects them from harm or danger. The ceremony ends with the tying of yellow coloured strings on the wrist, a symbolic notion of the presence of the Sangha in the laity. Of course, if one wishes, they can also make a donation to the monk concerned for his service by putting money into a metal box.

Lastly, to answer your question:

OP: Is Buddhism free from all superstitious blind beliefs or it is just an simple offshoot of Hinduism / Brahmanism ?

If by Buddhism, you are referring to what the Buddha taught, then the following passage from DN 2 would be self-explanatory. You might also be interested in "Is Buddhism just Hinduism stripped for export?".

"Whereas some contemplatives & brahmans, living off food given in faith, maintain themselves by wrong livelihood, by such "animal" arts as: reading marks on the limbs [e.g., palmistry]; reading omens and signs; interpreting celestial events [falling stars, comets]; interpreting dreams; reading features of the body [e.g., phrenology]; reading marks on cloth gnawed by mice; offering fire oblations, oblations from a ladle, oblations of husks, rice powder, rice grains, ghee, and oil; offering oblations from the mouth; offering blood-sacrifices; making predictions based on the fingertips; geomancy; making predictions for state officials; laying demons in a cemetery; placing spells on spirits; earth-skills [divining water and gems?]; snake-skills, poison-skills, scorpion-skills, rat-skills, bird-skills, crow-skills; predicting life spans; giving protective charms; casting horoscopes — he abstains from wrong livelihood, from "animal" arts such as these.

"Whereas some contemplatives & brahmans, living off food given in faith, maintain themselves by wrong livelihood, by such "animal" arts as: determining lucky and unlucky gems, staffs, garments, swords, arrows, bows, and other weapons; women, men, boys, girls, male slaves, female slaves; elephants, horses, buffaloes, bulls, cows, goats, rams, fowl, quails, lizards, rabbits, tortoises, and other animals — he abstains from wrong livelihood, from "animal" arts such as these.

"Whereas some contemplatives & brahmans, living off food given in faith, maintain themselves by wrong livelihood, by such "animal" arts as [forecasting]: the rulers will march forth; the rulers will not march forth; our rulers will attack, and their rulers will retreat; their rulers will attack, and our rulers will retreat; there will be triumph for our rulers and defeat for their rulers; there will be triumph for their rulers and defeat for our rulers; thus there will be triumph this one, defeat for that one — he abstains from wrong livelihood, from "animal" arts such as these.

"Whereas some contemplatives & brahmans, living off food given in faith, maintain themselves by wrong livelihood, by such "animal" arts as [forecasting]: there will be a lunar eclipse; there will be a solar eclipse; there will be an occultation of [a conjunction of the moon or a planet with] an asterism; the sun and moon will be favorable; the sun and moon will be unfavorable; the asterisms will be favorable; the asterisms will be unfavorable; there will be a meteor shower; there will be a flickering light on the horizon [an aurora?]; there will be an earthquake; there will be thunder coming from dry clouds; there will be a rising, a setting, a darkening, a brightening of the sun, moon, and asterisms; such will be the result of the lunar eclipse ... the rising, setting, darkening, brightening of the sun, moon, and asterisms — he abstains from wrong livelihood, from "animal" arts such as these.

"Whereas some contemplatives & brahmans, living off food given in faith, maintain themselves by wrong livelihood, by such "animal" arts as [forecasting]: there will be abundant rain; there will be a drought; there will be plenty; there will be famine; there will be rest and security; there will be danger; there will be disease; there will be freedom from disease; or they earn their living by accounting, counting, calculation, composing poetry, or teaching hedonistic arts and doctrines [lokāyata] — he abstains from wrong livelihood, from "animal" arts such as these.

"Whereas some contemplatives & brahmans, living off food given in faith, maintain themselves by wrong livelihood, by such "animal" arts as: calculating auspicious dates for marriages — both those in which the bride is brought home and those in which she is sent out; calculating auspicious dates for betrothals and divorces; for collecting debts or making investments and loans; reciting charms to make people attractive or unattractive; curing women who have undergone miscarriages or abortions; reciting spells to bind a man's tongue, to paralyze his jaws, to make him lose control over his hands, or to bring on deafness; getting oracular answers to questions addressed to a spirit in a mirror, in a young girl, or to a spirit medium; worshipping the sun, worshipping the Great Brahmā, bringing forth flames from the mouth, invoking the goddess of luck — he abstains from wrong livelihood, from "animal" arts such as these.

"Whereas some contemplatives & brahmans, living off food given in faith, maintain themselves by wrong livelihood, by such "animal" arts as: promising gifts to deities in return for favors; fulfilling such promises; demonology; reciting spells in earth houses [see earth skills, above]; inducing virility and impotence; preparing sites for construction; consecrating sites for construction; giving ceremonial mouthwashes and ceremonial baths; offering sacrificial fires; administering emetics, purges, purges from above, purges from below, head-purges; ear-oil, eye-drops, treatments through the nose, ointments, and counter-ointments; practicing eye-surgery [or: extractive surgery], general surgery, pediatrics; administering root-medicines and binding medicinal herbs — he abstains from wrong livelihood, from "animal" arts such as these. This, too, is part of his virtue."

"A monk thus consummate in virtue sees no danger anywhere from his restraint through virtue. Just as a head-anointed noble warrior king who has defeated his enemies sees no danger anywhere from his enemies, in the same way the monk thus consummate in virtue sees no danger anywhere from his restraint through virtue. Endowed with this noble aggregate of virtue, he is inwardly sensitive to the pleasure of being blameless. This is how a monk is consummate in virtue.

Our actions often have symbolic value. For example, when you say "good morning" to your neighbor - the morning may not be very good, it may be too hot or heavy raining, or your neighbor may be in bad mood - but we make this gesture for its symbolic value. We say good morning as a way to create positive emotional energy for the entire day. We are sending a message of hope and optimism to each other. We are wishing that good morning will lead to good day.

Or we shake hands. It may seem like a useless act, but in fact it's a symbol of peaceful contact. We use this gesture for its symbolic value. We are sending a message of trust and no-hate, no-fear to each other.

Similarly, some activities performed by the public figures are done as symbolic acts that create certain emotional energy in the people who participates, watches, or hears about them. They may send a message of respect, or compassion, or friendship etc. Just like good morning or hand-shaking, except it works at the national and international level.

Public prayers are just like that, they are symbolic actions that send a message of peace, hope, compassion or whatever the topic of the prayers. In Buddhism, many schools like Theravada, Mahayana, Tibetan Vajrayana - use the ceremony of offering, ceremony of puja, and public prayers - as a way to send a positive public message, that will create a certain positive mindset in the people. It may shift the mood of every single person very slightly, but on the overall scale it may have a pretty noticeable positive effect.

Maybe it's a politic and skilful thing to do -- analogous to the Prime Minister of New Zealand wearing a headscarf after the Christchurch mosque attack.

New Zealand PM Jacinda Ardern, wearing hijab, meets Christchurch terror attack victims' families

Perhaps it helps to represent that (the population of) Sri Lanka is Buddhist and Hindu too, and that populations (within countries and between) can co-exist peacefully.

That's just a guess though, I haven't read any publicity about his motive.