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Belief in God (called "faith" in Christianity) is not something that we manufacture within ourselves. It is a gift that comes from God. But the Christian God is the God who knows every thought, who hears every word, and who sees everything we do (Psalm 139). He is the God who created everyone, whether the person is Christian or not. Therefore, regardless of the words we use, if we direct our words to the Creator God, He can hear us when we ask for faith.
Furthermore, the Christian God also said that whoever calls on the name of the LORD will be saved (Joel 2:28-32, quoted by Peter in Acts 2:21 and by Paul in Romans 10:13). But what does it mean to call on the name of the LORD? In the context of those passages (Acts 2:14-41 and Romans 10:5-13) it is anyone who realizes that we have problems loving others the way we are supposed to despite how hard we have tried, and that we need help from the source of love: God himself. This is God as Christians understand Him, which is not 100% the same as the God of other religions. This God sent His Son Jesus who gave His life for us. Calling on the name of the LORD means trusting that this Jesus will help us love God and others, so our lives are not in conflict inside anymore.
You asked: Is there a prayer to God for the supplicant to believe in Him? As long as we genuinely surrender ourselves to Jesus and call Him Lord and plead with Him to fill us with his love so we can love others, it doesn't matter what words we use. An example:
Dear God. I don't know what to say. I just want to confess that I did many bad things in my life and that I am sorry. I don't want to live the way I used to anymore: self centered instead of wanting to please you. Please heal me from my wicked tendencies. Fill me with your love so I can love others like your Son Jesus. Help me to trust you above anyone else in the world. Open the door for me to learn about you by leading me to a trustworthy Christian church. In Jesus's name, I pray. Amen.
Prayer to God in Christianity is mostly like a person to person communication, like talking to another human being, although in this case, we are talking to God. It may be surprising to people coming from Hinduism / Buddhism, but in most types of Christian prayer no specific mental state is necessary (per Psalm 139 referred to above).
In this person to person prayer, we can simply visualize ourselves talking to the person of Jesus, who once walked on earth as a human being just like we are today, except that He is also God and now is accessible to us in our hearts. Again, as in Psalm 139, we have a basis to visualize Jesus as always being near us. We can talk anytime and anywhere to Him (doesn't have to be in a church), and we can listen to Him by reading His Word in the Bible and let the message there touch our hearts. So I would encourage that in conjunction to talking to Him, you would start reading the Bible, preferably with a trusted Christian friend / pastor to orient you.
There IS a mystical tradition in Christianity where the prayer is rather different and has some similarity with meditative prayer in other religion. But this is for Christians who are already advanced spiritually, who already started their journey with Jesus for quite sometime.
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Is there a prayer to God for...
Christians are not restricted to pre-written prayers. While some denominations have pre-written prayers, and extensively use them, even in those denominations the Christian is not prevented from praying in their own words for something they want and which they believe God would want for them. For other denominations prayer in your own words is expected and preferred. The same applies to people who do not call themselves Christians but want to ask God for something.
Is there a prayer to God for the supplicant to believe in Him?
I don't know of any pre-written prayer specifically asking for this. As I said above, that does not prevent any person from asking God for faith to believe in him.
Can someone ask God for faith to believe in him?
Yes, asking for faith is supported by the church and in the Bible. This applies to both Christians and not. Many Christians testify that before they believed in God they asked him to show himself to them. In particular there is the Biblical story of a man whose son was sick/possessed by a spirit and who prayed βI do believe; help me overcome my unbelief!β (Mark 9:24). Specifically he acknowledges that belief is not a simple thing, and he has both belief (or he would not be asking) and unbelief.