My problem is that I feel that every act of worship that I perform is done for the sake of having my prayer answered. I fast because the supplication of the fasting person will be answered; I pray the supererogatory prayers in order to come nearer to Allaah so that He would accept my supplication. In brief, I feel that all my worship is not purely for the sake of Allaah, and this feeling suffocates me. What should I do? Or is this ok?

All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger. 

This feeling is nothing but Waswaas (obsessive doubts), so you have to repel them and remain conscious of being sincere to Allah in all your acts and avoid illusions and doubts in this field as they are from the devil in order to stop you from performing acts of worship and put a barrier between you and them.

Moreover, performing an act of worship in order to get closer to Allah with the intention to make it a cause for having your prayer answered does not undermine sincerity, and it does not invalidate your acts of worship.

Allah knows best.