  In this work we study the generalized Weierstrass semigroup $\widehat{H}
(\mathbf{P}_m)$ at an $m$-tuple $\mathbf{P}_m = (P_{1}, \ldots , P_{m})$ of
rational points on certain curves admitting a plane model of the form $f(y) =
g(x)$ over $\mathbb{F}_{q}$, where ${f(T),g(T)\in \mathbb{F}_q[T]}$. In
particular, we compute the generating set $\widehat{\Gamma}(\mathbf{P}_m)$ of
$\widehat{H} (\mathbf{P}_m)$ and, as a consequence, we explicit a basis for
Riemann-Roch spaces of divisors with support in $\{P_{1}, \ldots , P_{m}\}$ on
these curves, generalizing results of Maharaj, Matthews, and Pirsic.
