Does a person become a polytheist (mushrik) due to the polytheism of intention (shirk niyyah) if he originally intended an act for worldly gain, and then intended it to be for the sake of Allah during its performance? And what should he do to make his act purely for Allah?

Visiting the sick does not lack validity without intention, but it does lack the entitlement to reward. So, whoever does not intend to draw closer to Allah by it does not sin, but misses out on the reward. As for ordinary actions, like visiting the sick, they do not become acts of worship and do not entail reward except with the intention of obeying Allah. What helps with sincerity is observing Allah, being mindful of His awareness, and reflecting on the religious texts that encourage sincerity. As for whispers (waswas), one must ignore them and not pay attention to them. If they relate to creed, one should say, "I believe in Allah," then stop and seek refuge in Allah from the accursed devil.