Final answer:
Thomas Becket is appointed as the Archbishop of Canterbury by King Henry II, a position he does not want because he foresees conflicting loyalties between the church and the king, leading to his eventual martyrdom.
Step-by-step explanation:
The position that King Henry II gives to Thomas Becket which Becket is reluctant to accept is that of Archbishop of Canterbury. Becket fears this appointment because he anticipates that it will lead to a conflict of interest between his duties to the church and his allegiance to the King. Becket's apprehension is well-founded, as his later actions as Archbishop do indeed lead to a significant rift between him and the King, ultimately resulting in Becket's martyrdom.