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We present improved approximation algorithms for some problems in the related areas of Capacitated Network Design and Flexible Graph Connectivity. In the Cap-k-ECSS problem, we are given a graph G = (V,E) whose edges have non-negative costs and positive integer capacities, and the goal is to find a minimum-cost edge-set F such that every non-trivial cut of the graph G' = (V,F) has capacity at least k. Let n = |V| and let u_{min} (respectively, u_{max}) denote the minimum (respectively, maximum) capacity of an edge; assume that u_{max} ≤ k. We present an O(log({k}/u_{min}))-approximation algorithm for the Cap-k-ECSS problem, asymptotically improving upon the previous best approximation ratio of min(O(log{n}), k, 2u_{max}, 6 ⋅ {⌈ k/u_{min} ⌉}) whenever log(k/u_{min}) = o(log{n}) and u_{max} is sufficiently large. 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 edge-set F such that the subgraph G' = (V, F) remains p-edge connected upon removal of any q unsafe edges from F. We present an 8-approximation algorithm for the (1,q)-FGC problem that improves upon the previous best approximation ratio of (q+1). Both of our results are obtained by using natural LP relaxations strengthened with the knapsack-cover inequalities, and then, during the rounding process, utilizing a recent O(1)-approximation algorithm for the Cover Small Cuts problem. In the latter problem, the goal is to find a minimum-cost set of links such that each non-trivial cut of capacity less than a specified value is covered by a link. We also show that the problem of covering small cuts inherently arises in another variant of (p,q)-FGC. Specifically, we give Cook reductions that preserve approximation ratios within O(1) factors between the (2,q)-FGC problem and the 2-Cover Small Cuts problem; in the latter problem, each small cut needs to be covered by two links.