Is it known whether hypergraph minimal covers are P-enumerable?
You are looking for an output-polynomial algorithm for enumerating minimal transversals of hypergraphs (or hitting sets for set systems). According to Golovach et al. (ICALP 2013),


  The question whether the minimal transversals
  of a hypergraph can be enumerated in output polynomial time is a fundamental and
  challenging question; it has been open for several decades and has triggered extensive
  research. [...]
  
  Whether or not all minimal transversals of a
  hypergraph can be listed in output polynomial time has been identied as a fundamental
  challenge in a long list of seminal papers, e.g., [10, 11, 12, 13, 14, 19, 30], and it remains
  unresolved despite continuous attempts since the 1980's.