Let $R$ be a commutative Noetherian ring and ${\mathfrak a}$ an ideal of $R$. We introduce the concept of ${\mathfrak a}$-weakly Laskerian $R$-modules, and we show that if $M$ is an ${\mathfrak a}$-weakly Laskerian $R$-module and $s$ is a non-negative integer such that ${\rm Ext}^j_R(R/{\mathfrak a}, H^i_{{\mathfrak a}}(M))$ is ${\mathfrak a}$-weakly Laskerian for all $i<s$ and all $j$, then for any ${\mathfrak a}$-weakly Laskerian submodule $X$ of $H^s_{{\mathfrak a}}(M)$, the $R$-module ${\rm Hom}_R(R/{\mathfrak a},H^s_{{\mathfrak a}}(M)/X)$ is ${\mathfrak a}$-weakly Laskerian. In particular, the set of associated primes of $H^s_{\mathfrak a}(M)/X$ is finite. As a consequence, it follows that if $M$ is a finitely generated $R$-module and $N$ is an ${\mathfrak a}$-weakly Laskerian $R$-module such that $ H^i_{{\mathfrak a}}(N)$ is ${\mathfrak a}$-weakly Laskerian for all $i<s$, then the set of associated primes of $H^s_{\mathfrak a}(M, N)$ is finite. This generalizes the main result of S. Sohrabi Laleh, M. Y. Sadeghi, and M. Hanifi Mostaghim (2012).
Let $R$ be a commutative Noetherian ring with identity and $I$ an ideal of $R$. It is shown that, if $M$ is a non-zero minimax $R$-module such that $\dim \mathop {\rm Supp} H^i_I (M) \leq 1$ for all $i$, then the $R$-module $H^i_I(M)$ is $I$-cominimax for all $i$. In fact, $H^i_I(M)$ is $I$-cofinite for all $i\geq 1$. Also, we prove that for a weakly Laskerian $R$-module $M$, if $R$ is local and $t$ is a non-negative integer such that $\dim \mathop {\rm Supp} H^i_I (M)\leq 2$ for all $i<t$, then ${\rm Ext}^j_R (R/I, H^i_I (M))$ and ${\rm Hom}_R(R/I, H^t_I(M))$ are weakly Laskerian for all $i<t$ and all $j \geq 0$. As a consequence, the set of associated primes of $H^i_I (M)$ is finite for all $i\geq 0$, whenever $\dim R/I \leq 2$ and $M$ is weakly Laskerian.