In this paper, we study some subvarieties of a semiring variety determined by certain additional identities. We first present alternative characterizations for equivalences $\stackrel{+}{\mathcal{H}}\cap \stackrel{\cdot }{\mathcal{L}}$, $\stackrel{+}{\mathcal{H}}\cap \stackrel{\cdot }{\mathcal{R}}$, $\stackrel{+}{\mathcal{H}}\cap \stackrel{\cdot }{\mathcal{D}}$, $\stackrel{+}{\mathcal{H}}\vee \stackrel{\cdot }{\mathcal{L}}$, $\stackrel{+}{\mathcal{H}}\vee \stackrel{\cdot }{\mathcal{R}}$, $\stackrel{+}{\mathcal{H}}\vee \stackrel{\cdot }{\mathcal{D}}$. Then we give the sufficient and necessary conditions for these equivalences to be congruence. Finally, we prove that semiring classes defined by these congruences are varieties and provide equational bases.