Alternately, for any such $\Gamma$ and $C$, we always have a short exact sequence $$0 \to O_{\Gamma} \to f_* O_C \to F \to 0,$$ where $F$ is a finite length $\mathcal{O}_{\Gamma}$-module supported where $\Gamma$ is non-smooth.

Twisting by a very positive Cartier divisor $L$ and taking cohomology we get $$ 0 \to H^0(\Gamma, O_{\Gamma}(L)) \to H^0(C, O_{C}(f^*L)) \to H^0(\Gamma, F(L)) \to H^1(\Gamma, O_{\Gamma}(L))$$

Because $L$ is very very ample, $H^0(\Gamma, F(L)) \neq 0$ but $H^1(\Gamma, O_{\Gamma}(L)) = 0$ by Serre vanishing. And so you have a counter-example.

Alternately, for any such $\Gamma$ and $C$, we always have a short exact sequence $$0 \to O_{\Gamma} \to f_* O_C \to F \to 0,$$ where $F$ is a finite length $\mathcal{O}_{\Gamma}$-module supported where $\Gamma$ is non-smooth.

Twisting by a very positive (ie, high multiple of an ample) Cartier divisor $L$ and taking cohomology we get $$ 0 \to H^0(\Gamma, O_{\Gamma}(L)) \to H^0(C, O_{C}(f^*L)) \to H^0(\Gamma, F(L)) \to H^1(\Gamma, O_{\Gamma}(L))$$

Now, $H^0(\Gamma, F(L)) \neq 0$ but $H^1(\Gamma, O_{\Gamma}(L)) = 0$ by Serre vanishing. And so you have a counter-example.