Question

Why doesn't king Arthur trust Mordred?

Answer 1

Although King Arthur initially tried to kill Mordred because of his embarrassment concerning Mordred's incestuous birth, he eventually trusts him enough to leave his kingdom in Mordred's care when he goes to battle Lancelot.

Answer 2

Arthur did trust Mordred. Arthur made Mordred regent over Britain when Arthur went to Gaul to make war against Lancelot.

Then, while Arthur was in Gaul, Mordred rebelled against Arthur and tried to make Guinevere his wife. At that point Arthur quite reasonably stopped trusting him.