Cork has already missed a large chunk of the season following a fractured foot and he suffered a knock in Saturday's 1-1 Premier League draw against Brighton, so his fitness will be checked.
The quartet were absent for last Wednesday's defeat to Manchester City as well as the weekend stalemate against Brighton, and Clarets manager Sean Dyche is reluctant to rush anyone back.
Bournemouth top goalscorer Dominic Solanke will miss the midweek trip to Turf Moor with an ankle injury and could be set for a spell on the sidelines with the problem.
Fellow forward Shane Long, who made his Cherries debut on Saturday, is also absent due to being cup-tied after he appeared in the competition with Southampton last month.
It means caretaker boss Jonathan Woodgate could turn to Sam Surridge, while Jack Wilshere will aim to build on his goal against Birmingham last weekend and he also netted in the previous round victory over Crawley.