In cyclic process, the system returns to its initial state as shown below, so ΔU=0.
However, reversibility is a hypothetical concept which means a process is reversible, if it can be turned back such that both the system and surroundings return to their original state, with no other change anywhere else in universe. In cyclic process A→B→C→D→A, the system returns back its temperature, pressure, volume at A its (initial state) but some work is done on system (in given figure) and system releases heat. Work done on the system is not reversed back (i.e. system has not done work on surroundings). Thus, reversibility has nothing to do with cyclic process.
Therefore, Assertion and Reason are correct but Reason is not the correct explanation of Assertion.