It is a classical problem in 4-manifold topology to get lower bounds on the genus of an embedded surface \(\Sigma^2 \hookrightarrow X^4\) in terms of its homology class \([\Sigma^2] \in H_2 (X^4; \mathbb{Z})\). As a famous example proved by [P. B. Kronheimer and T. S. Mrowka [Math. Res. Lett. 1, No. 6, 797-808 (1994; Zbl 0851.57023)] using the Seiberg-Witten equations, the classical Thom conjecture states that \(g(\Sigma^2) \geq {1\over 2} (d-1) (d-2)\) for \(\Sigma^2 \hookrightarrow \mathbb{C} P^2\) representing \([\Sigma^2] =d[\mathbb{C} P^1] \) with \(d>0\). Before the introduction of gauge theoretical methods genus bounds were obtained by assuming divisibility conditions on the class \([\Sigma^2]\) and studying the associated branched cover of \(X^4\). In this short note the author combines this classical technique with Furuta’s \({10 \over 8}\)-theorem to obtain the following genus bound: \[ g (\Sigma^2) \geq 1+ {5n (1+q) \over 24q} +{2\over q-1} -{q\over 2 (q-1)} \left (e(X^4)+ {5\over 4} \sigma (X^4) \right), \] where \(q>1\) is a prime power dividing \([\Sigma^2]\), \(n\) denotes the self-intersection of \(\Sigma^2\), and \(e(X^4)\), \(\sigma (X^4)\) are the Euler characteristic and signature of \(X^4\), respectively. This bound holds under the assumption that the associated branched cover of \(X^4\) is spin.