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

Generalized Weierstrass semigroups and Riemann-Roch spaces for certain curves with separated variables