Tottenham - who are fifth in the Premier League table - face a midweek trip to Blackburn, followed by a home game with Bolton and an away game at Sunderland on the 12 February.
Having seen a potential route to silverware permanently blocked via Sunday's heavy defeat at Fulham, Spurs will be even more determined to ensure they secure a second-successive Champions League qualification via a top-four league finish.
The north Londoners were uncharacteristically sloppy in the Cup tie at Craven Cottage, where two Danny Murphy penalties in the opening 15 minutes set the tone for the match.
The first was awarded for Alan Hutton's foul on Clint Dempsey and the second for Dawson's dragging back of Moussa Dembele when the striker had a clean shooting opportunity.
Redknapp suggested Dawson's dismissal was "harsh", but did not shy away from blaming his players for the mistakes that led to the goals.
"We got off to a disastrous start and it was game over after the two penalties. They were uncharacteristic errors from people," Redknapp reflected.
"You don't see Michael making too many errors but he has had a nightmare start, giving the ball away, the penalty and then the sending-off.
"It has nothing to do with tactics when someone gives the ball away and we bring them down for a penalty, lose the ball and try to dribble out of trouble."
Fulham killed the tie before half time courtesy of Brede Hangeland's close-range finish from a corner in the 23rd minute and Dembele's superb solo effort seconds before the break.
The second half was a mere formality, although it could have been worse for the visitors had Aaron Hughes' header not struck the bar and Jermaine Jenas clearing off the line from Hangeland as he pounced on the rebound.
"I think it's been coming, especially against Spurs," Fulham manager Mark Hughes commented. "We've played well on two occasions against them and haven't had any reward.
Murphy's goals were his first since March last year
"We were on the front foot, forced errors from their back four."
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