(A very partial answer only, but too long for a comment.)

In light of your second comment under Monroe's answer, I interpret your question as follows:

Given a formula $\varphi(x)$, write $\kappa_\varphi$ for the least cardinal satisfying $\varphi$. ($\kappa_\varphi$ could be undefined, of course. But you are only concerned with formulas which are satisfied by a unique cardinal.)

Write $C(\kappa)$ to abbreviate $\kappa^{cf(\kappa)} < \kappa^{\kappa}$.

How can we find out if "ZFC + $\kappa_\varphi$ exists + $C(\kappa_\varphi)$" is consistent?

A sufficient condition is the following: $\varphi$ is absolute under cardinal-preserving extensions (so in particular, $\varphi$ may use ordinal arithmetic, cardinal successor, the $\aleph$-function, etc), ZFC proves that $\kappa_\varphi$ exists and is singular. (This is really just a reformulation of Monroe Eskew's answer.)

An example is the formula $\kappa=\aleph_\omega$.