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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 theuniqueness of critical points of energy function and the injectivity of Hopfdifferential if $u_0$ is a covering map. On the other hand, if $u_0$ is abranched covering, we show that the uniqueness of critical points fails if$u_0$ is a non-simple branched covering, and prove the injectivity of Hopfdifferential $\Phi:\mc{T}(S)\to \op{QD}(\Sigma,g)$ when $g=[u_0^* h]$ for somehyperbolic metric $h$ on $S$.

Harmonic maps between surfaces homotopic to a (branched) covering map