In the paper, we consider the harmonic maps between surfaces \(\Sigma \) and S in the homotopy class of a (branched) covering map \(u_0\) . We prove the uniqueness of critical points of the energy function and the injectivity of the Hopf differential map if \(u_0\) is a covering map. On the other hand, if \(u_0\) is a branched covering, we show that the uniqueness of critical points fails if \(u_0\) is a non-simple branched covering and prove the injectivity of Hopf differential map \(\Phi :\mathcal {T}(S)\rightarrow \operatorname {QD}(\Sigma ,g)\) when \(g=[u_0^* h]\) for some hyperbolic metric h on S. This provides concrete counterexamples to the non-uniqueness of critical points in the branch covering case.
