Cleaves peptides in various proteins in a process that requires ATP hydrolysis. Has a limited peptidase activity in the absence of ATP-binding subunits ClpC, ClpE or ClpX. Has a chymotrypsin-like activity. Plays a major role in the degradation of misfolded proteins. ClpXP is involved in the complete degradation of the site-2 clipped anti-sigma-W factor RsiW. This results in the release of SigW and the transcriptional activation of genes under the control of the sigma-W factor. Probably the major protease that degrades proteins tagged by trans-translation