As a first-round answer and some detailed about this problem:

Let $G$ be a $p-$group of odd order such that every abelian normal subgroup has at most $k$ generators, then every subgroup of $G$ has at most $C(k+1,2)$ generators. If $p>2$ and $k=1$, it results that $G$ is cyclic.

In the thesis "Abelian subgroups of $p-$groups" The thesis by Soo-Seng Siah, this problem is studied and is studied with related to the term of "depth" and "normal-depth" of a $p-$group $G$. You can see the section 4 of this thesis for some more information.

It seems that in general the question is so hard and there is not a general classification.

As a first-round answer and some detailed about this problem:

Let $G$ be a $p-$group of odd order such that every abelian normal subgroup has at most $k$ generators, then every subgroup of $G$ has at most $C(k+1,2)$ generators.
For your question, if we have $p>2$ and $k=1$, it is a classical result that $G$ is cyclic; see the thesis which I introduced below.

In the thesis "Abelian subgroups of $p-$groups" The thesis by Soo-Seng Siah, this problem and its generalization is studied with related to the term of "depth" and "normal-depth" of a $p-$group $G$. You can see the section 4 of this thesis for some more information.