We present improved approximation algorithms for some problems in the related areas of Flexible Graph Connectivity and Capacitated Network Design. In the $(p,q)$-Flexible Graph Connectivity problem, denoted $(p,q)$-FGC, the input is a graph $G(V, E)$ where $E$ is partitioned into safe and unsafe edges, and the goal is to find a minimum cost set of edges $F$ such that the subgraph $G'(V, F)$ remains $p$-edge connected upon removal of any $q$ unsafe edges from $F$. In the related Cap-$k$-ECSS problem, we are given a graph $G(V,E)$ whose edges have arbitrary integer capacities, and the goal is to find a minimum cost subset of edges $F$ such that the graph $G'(V,F)$ is $k$-edge connected. We obtain a $7$-approximation algorithm for the $(1,q)$-FGC problem that improves upon the previous best $(q+1)$-approximation. We also give an $O(\log{k})$-approximation algorithm for the Cap-$k$-ECSS problem, improving upon the previous best $O(\log{n})$-approximation whenever $k = o(n)$. Both these results are obtained by using natural LP relaxations strengthened with the knapsack-cover inequalities, and then during the rounding process utilizing an $O(1)$-approximation algorithm for the problem of covering small cuts. We also show that the the problem of covering small cuts inherently arises in another variant of $(p,q)$-FGC. Specifically, we show $O(1)$-approximate reductions between the $(2,q)$-FGC problem and the 2-Cover$\;$Small$\;$Cuts problem where each small cut needs to be covered twice.