The striker has yet to feature since the restart due to a hip injury but is now back in training.
Cork may need surgery on an ankle complaint and he and Mee join Matt Lowton and Ashley Barnes on the sidelines. However, Chris Wood and Johann Berg Gudmundsson made cameos off the bench against Sheffield United, while Robbie Brady was an unused substitute after recent injuries and the trio will feature in the squad again.
West Ham provisional squad: Fabianski, Fredericks, Diop, Ogbonna, Cresswell, Rice, Soucek, Lanzini, Bowen, Fornals, Antonio, Randolph, Johnson, Balbuena, Noble, Wilshere, Yarmolenko, Haller, Ajeti, Silva.
Burnley provisional squad: Pope, Peacock-Farrell, Jensen, Taylor, Tarkowski, Pieters, Bardsley, Long, Dunne, Thomas, Gudmundsson, Brownhill, McNeil, Brady, Westwood, Goodridge, Rodriguez, Vydra, Thompson, Wood, Benson.