We prove that, for any $tge 3$, there exists a constant $c=c(t)>0$ such that any $d$-regular $n$-vertex graph with the second largest eigenvalue in absolute value~$lambda$ satisfying $lambdale c d^{t-1}/n^{t-2}$ contains vertex-disjoint copies of $K_t$ covering all but at most $n^{1-1/(8t^4)}$ vertices. This provides further support for the conjecture of Krivelevich, Sudakov and Szabo [emph{Triangle factors in sparse pseudo-random graphs}, Combinatorica textbf{24} (2004), pp.~403--426] that $(n,d,lambda)$-graphs with $nin 3mathbb{N}$ and $lambdaleq cd^{2}/n$ for a suitably small absolute constant~$c>0$ contain triangle-factors.
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