Carroll has endured a difficult time at Upton Park since making his loan spell permanent in June 2013 and now faces another four months on the sidelines as he requires surgery on an ankle complaint.
The England international made just 15 Premier League appearances last term and Sullivan, who accepts that spending £15m on the centre forward may have been a mistake, knows that reinforcements are needed in his absence.
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"You can never have enough strikers and really, with Andy gone, we haven't got enough.
"It was desperate last year," Sullivan added of Carroll's struggles for fitness. "We probably made a mistake by putting all our eggs in one basket.
"Had we known what we know now, we wouldn't have signed the player but it's easy to be wise after the event.
"We were assured by our physio - who has since left the club - that the player would be available for the start of the season [but] he wasn't available until January or February. A mistake was made and we were given the wrong information. We based our decision and our purchase of the player on what the medical department told us."
West Ham have already recruited forwards Enner Valencia and Mauro Zarate so far this summer and winger Stewart Downing, while saddened for his team-mate, sees it as a chance for the new faces to push for a start.
"[Carroll] was really looking forward to getting up and running this season and he's had a bit of a setback. It's disappointing for him," Downing told the media.
"It's a big blow for the team. He's a big player for us but it's a chance for someone else to come in and stake their claim. It's disappointing for Andy but it's a long season and we'll get him back as quick as we can."