The Cottagers back-line looked desperate in the home defeat to Villa.
"I apologise to Fulham's supporters for our performance tonight," Khan said.
"We've looked to add centre-backs since Wembley, I'm sorry we haven't yet as two got COVID and we lost a Free we thought was close and had another issue with a fourth centre-back.
"I promise players in and better efforts from this squad.
"I should and will apologise repeatedly for that performance. I'm sorry everybody. We all need to do a better job.
"Everyone at the club for the past year worked really hard to get the team up, & now we need to work significantly harder to stay up. I promise better efforts than today."