West Ham have surpassed expectations in the Premier League this season and are seventh in the table with 32 points from 20 matches, but the high-flying Hammers picked up one point from three games over the festive period.
Allardyce is hoping for a return to form against an out-of-sorts Everton at Goodison Park on Tuesday and said the Hammers will be taking the FA Cup more seriously than last season when they were fighting for Premier League survival.
"Last year it was the bottom of our priorities because we were concentrating on getting out of trouble," Allardyce told a news conference on Monday.
"Looking at the team there won't be too many changes. We haven't had the results we wanted recently but I think the performances have still been very good.
"We'll play the strongest team available against Everton.
"It's not possible to play the best team each game because of the fatigue levels. But we've had four or five days rest which has helped so we can play a strong team."
League One (third tier) Sheffield United provided the only major shock of the third-round when they beat Premier League Queens Park Rangers 3-0 on Sunday.
Despite a lack of giant-killings, the 60-year-old Allardyce still has great affection for soccer's oldest domestic cup competition.
"It's a great tournament with the opportunity of playing at Wembley at the end," he said.
"I remember watching it from the age of about eight or nine and the chance of an underdog beating a big boy hasn't lost its glamour.
"There hasn't been so many shocks in this round but the glamour is still there."
(Reporting by Michael Hann, editing by Ed Osmond)
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