Fortunately, this has been lessened slightly with the loan signing of Wolves defender Roger Johnson.
The Hammers sit 19th in the Barclays Premier League and an inexperienced side were thrashed 5-0 by Nottingham Forest in the third round of the FA Cup on Sunday.
Despite booking a Capital One Cup semi-final with Manchester City, the first leg of which takes place at the Etihad on Wednesday, Allardyce's position has started to come under question following a run of one win in13 league outings.
But Cottee believes the long list of injured first-team regulars has skewed the opinions on both the players and manager at Upton Park and feels Allardyce deserves everyone's backing.
"It is hard to judge Sam as a manager at the moment," Cottee told Press Association Sport.
"His best striker (Carroll) has been out all season and his best centre-half, Winston Reid, is missing now.
"It is hard to judge whether we are going in the right or wrong direction overall.
"But in the short term, it is important that everyone gets behind the manager and that above all else, the players put in performances.
"Sam can do a Winston Churchill speech before a game, but if the players don't go out and perform, it is not going to make any difference."
The club's co-chairmen issued an open letter to West Ham fans on Tuesday in which they also expressed their desire to stand by Allardyce - and promised to support their manager in the January transfer window.
"We know Sam has not lost his ambition or desire and is committed to making West Ham United a great Premier League club," the letter read.
"Sam has been handed a near impossible task of coming through three crucial games in the space of six days with a squad of only 14 fit and available senior players.
"We have experienced difficult times before and we know what will get us results - hard work, determination and perseverance.
"To that end, we are all working round the clock to do everything in our powers to help arrest our slump in form.
"January is not an easy time to do business but we are pleased to confirm we are close to securing one of our key striking targets. A defender is high on the list of priorities as well and progress is being made on that front."
It transpired that defence was the first area that was strengthened as Johnson, 30, arrived from Wolves on a loan deal until the end of the season.
The centre-back has played in the Premier League for both Wolves and Birmingham and goes straight in to Allardyce's squad for the semi-final at Manchester City.
Monaco forward Lacina Traore continues to be linked with a similar loan move as the club look to get moving after being turned down by Everton defender John Heitinga at the start of the window.