What are examples of 4-manifolds that have more Morse critical points than topologically required? For example, $\pi_1(M), H_*(M; \mathbb{Z})$ have less than $k$ generators but any Morse function on $M$ has more than $k$ critical points.
Edit PVAL-inactive explains in the comments that in dimension 4, it is a big open problem whether simply-connected manifolds admit perfect Morse functions and so now looking at non-simply-connected manifold looks more promising.
Also, what are some 4-manifolds $M$ such that the unit disk tangent bundle $DTM$ requires fewer critical points than $M$? A related question: what are some 4-manifolds $M,N$ that are not diffeomorphic but the unit disk tangent bundles $DTM, DTN$ are diffeomorphic?
Maybe there are examples of some of theses things in dimension 3?
It's a well known open question whether every smooth closed simply-connected smooth 4-manifold admits a so-called perfect Morse function (a Morse function with only critical points of even index), so you are unlikely to find a compact simply-connected examples in your first question.
In the non-compact case, examples are pretty plentiful. First of all, there are plenty of contractible 4-manifolds which are not simply-connected at infinity. Many of these are so-called Mazur manifolds, which are the result of attaching a 2-handle to a winding number one knot in the boundary of $S^1 \times D^3$ (so these contractible manifolds admit Morse functions with two critical points).
There are also exotic examples. Due to a theorem of Taylor, any large exotic $\Bbb R^4$ (a manifold homeomorphic to $\Bbb R^4$ which does not smoothly embed in $\Bbb R^4$ has infinitely many index 3 critical points for any proper Morse function. As any contractible and simply-connected at infinity $n$-manifold for $n \ne 4$ is diffeomorphic to $\Bbb R^n$, the disk bundles of such exotic 4-manifolds will give examples to your second question (there should be loads of compact examples as well but not necessarily with differing Morse numbers).