Can we reduce an NP complete item to an NP item which is $\bf{non}$ P?
No, it is not possible to prove that all $NP$ problems which are not $P$ are $NP$-complete. In fact, the opposite has been shown: unless $P=NP$, there are $NP$ intermediate problems, neither in $P$, nor $NP$-complete. See https://en.wikipedia.org/wiki/NP-intermediate

So if you reduce an $NP$-complete problem to another $NP$ problem, all you've done is proved that that problem is also $NP$-complete.