Fewest induced subgraphs of order <= k which cover every edge
Your problem is known as $k$-clique cover. Goldschmidt, Hochbaum, Hurkens and Yu describe several approximation algorithms in their paper Approximation algorithms for the $k$-clique covering problem. The paper is from 1995, so there are likely newer works on this topic, which you can find by scanning the papers citing this paper.