I'm stuck with this exercise.
Let $$p : E \to X $$ be a covering map.
Y is a connected and locally path-connected topological space, $$f : Y \to X $$ is a continous map.
The claim is that $$f^*p : \lbrace (e,y) \in E \times Y | \ p(e) = f(y) \rbrace \to Y $$ $$f^*p(e,y) = y $$ is a covering map.
I suppose I have to use the fact that $$id \times p : Y \times E \to Y \times X $$ is a covering map.
How to proceed ?
For each $y\in Y$, some neighborhood $U$ of $f(y)=:x$ in $X$ has preimage in $E$ that is a disjoint union $E_1\cup E_2\cup \ldots$ of open subsets of $E$, each of them being homeomorphic to $U$. Some neighborhood $V\subseteq Y$ of $y$ is mapped to $f(V)\subseteq U$ by $f$. The preimage of $V$ in $\{(e,y):\,p(e)=f(y)\}$ is then a disjoint union of pieces $\{(e,y):\,e\in E_i,\,y\in V\}$, each of them homeomorphic to $V$.
This is a special case of the pullback of a fibre bundle (in your case the fibre is discrete).