Buddhist view on past lives influencing the present

The Buddha discouraged such ideas.

Having approached the brahmans & contemplatives who hold that... 'Whatever a person experiences... is all caused by what was done in the past,' I said to them: 'Is it true that you hold that... "Whatever a person experiences... is all caused by what was done in the past?"' Thus asked by me, they admitted, 'Yes.' Then I said to them, 'Then in that case, a person is a killer of living beings because of what was done in the past. A person is a thief... unchaste... a liar... a divisive speaker... a harsh speaker... an idle chatterer... greedy... malicious... a holder of wrong views because of what was done in the past.' When one falls back on what was done in the past as being essential, monks, there is no desire, no effort [at the thought], 'This should be done. This shouldn't be done.' When one can't pin down as a truth or reality what should & shouldn't be done, one dwells bewildered & unprotected. One cannot righteously refer to oneself as a contemplative. This was my first righteous refutation of those brahmans & contemplatives who hold to such teachings, such views. AN 3.61

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Now when those ascetics and brahmins hold such a doctrine and view as this, ‘Whatever a person experiences, whether it be pleasant or painful or neither-painful-nor-pleasant, all that is caused by what was done in the past,’ they overshoot what one knows by oneself and they overshoot what is considered to be true in the world. Therefore I say that this is wrong on the part of those ascetics and brahmins. SN 36.21

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Friend, the Blessed One has said that pleasure and pain are dependently arisen. Dependent on what? Dependent on contact. If one were to speak thus one would be stating what has been said by the Blessed One and would not misrepresent him with what is contrary to fact; one would explain in accordance with the Dhamma, and no reasonable consequence of one’s assertion would give ground for criticism.

Therein, friend, in the case of those ascetics and brahmins, proponents of kamma, who maintain that pleasure and pain are created by oneself, and those who maintain that pleasure and pain are created by another, and those who maintain that pleasure and pain are created both by oneself and by another, and those who maintain that pleasure and pain have arisen fortuitously, being created neither by oneself nor by another—in each case that is conditioned by contact.

Therein, friends, in the case of those ascetics and brahmins, proponents of kamma, who maintain that pleasure and pain are created by oneself, and those who maintain that pleasure and pain are created by another, and those who maintain that pleasure and pain are created both by oneself and by another, and those who maintain that pleasure and pain have arisen fortuitously, being created neither by oneself nor by another—in each case it is impossible that they will experience anything without contact.

SN 12.25

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Now what, monks, is old kamma? The eye ought to be viewed as old kamma, capable of being fabricated, willed & felt. The ear... The nose... The tongue... The body... The intellect ought to be viewed as old kamma, capable of being fabricated, willed & felt.. This is called old kamma. SN 35.145

Your question is not about whether past life actions influence present life circumstances. For that, see this answer and this sutta.

Rather, you are asking if your past life thoughts, affect your present life thoughts (or thought choices).

In a single lifetime, the answer is yes according to Dvedhavitakka Sutta.

"Whatever a monk keeps pursuing with his thinking & pondering, that becomes the inclination of his awareness. If a monk keeps pursuing thinking imbued with sensuality, abandoning thinking imbued with renunciation, his mind is bent by that thinking imbued with sensuality.

But does this cross into other lifetimes?

From the Nakhasikha Sutta, Samudda Sutta and Pokkharani Sutta:

"In the same way, monks, for a disciple of the noble ones who is consummate in view, an individual who has broken through [to stream-entry], the suffering & stress that is totally ended & extinguished is far greater. That which remains in the state of having at most seven remaining lifetimes is next to nothing: it's not a hundredth, a thousandth, a one hundred-thousandth, when compared with the previous mass of suffering. That's how great the benefit is of breaking through to the Dhamma, monks. That's how great the benefit is of obtaining the Dhamma eye."

When one obtains the Dhamma eye, and attains stream-entry, he would only have at most, seven remaining lifetimes. The Dhamma eye is explained as the ability to see firsthand and experientially, the impermanence of the five aggregates and dependent origination.

This is obviously an example in the Pali Canon that at least some thoughts or knowledge could cross over into the next life. In this case, it is the strong and irrevocable inclination towards the Dhamma in future lives, due to the opening of the Dhamma eye.

Also see this question and this answer.

Some say that child prodigies bring with them the strong inclinations of the immediate previous life, but I think that cannot be found in the scripture, and the Buddha would not bother to answer this (parable of the poisoned arrow, parable of the simsapa leaves, etc.).