We investigate Riemannian manifolds $(M^n,g)$ whose curvature operator of the second kind $\mathring{R}$ satisfies the condition \begin{equation*} α^{-1} (λ_1 +\cdots +λ_α) > - θ\barλ, \end{equation*} where $λ_1 \leq \cdots \leq λ_{(n-1)(n+2)/2}$ are the eigenvalues of $\mathring{R}$, $\barλ$ is their average, and $θ> -1$. Under such conditions with optimal $θ$ depending on $n$ and $α$, we prove two differentiable sphere theorems in dimensions three and four, a homological sphere theorem in higher dimensions, and a curvature characterization of Kähler space forms. These results generalize recent works corresponding to $θ=0$ of Cao-Gursky-Tran, Nienhaus-Petersen-Wink, and the author. Moreover, examples are provided to demonstrate the sharpness of all results.