Here is an inelegantly programmed solution, which can be streamlined, if you wish.
Here is a list of all strings of length exactly 6 that have exactly 4 Hs:
xxx = StringJoin /@
Select[Tuples[{"H", "T"}, 6], Counts[#]["H"] == 4 &]
(* {"HHHHTT", "HHHTHT", "HHHTTH", "HHTHHT", "HHTHTH", "HHTTHH", "HTHHHT", "HTHHTH", "HTHTHH", "HTTHHH", "THHHHT", "THHHTH", "THHTHH", "THTHHH", "TTHHHH"} *)
If the last letter in a string is a T, then that string is "useless" and should be deleted. Here's a set that has this property, i.e., is either a length-6 string ending in H or a length-5 string with exactly 4 Hs:
yyy = If[StringEndsQ[#, "T"], StringDrop[#, -1], #] & /@ xxx
(* {"HHHHT", "HHHTH", "HHHTTH", "HHTHH", "HHTHTH", "HHTTHH", "HTHHH", "HTHHTH", "HTHTHH", "HTTHHH", "THHHH", "THHHTH", "THHTHH", "THTHHH", "TTHHHH"} *)
If the last letter of a string in this list is a T, then that string is "useless" and should be deleted:
If[StringEndsQ[#, "T"], StringDrop[#, -1], #] & /@ yyy
(* {"HHHH", "HHHTH", "HHHTTH", "HHTHH", "HHTHTH", "HHTTHH", "HTHHH", "HTHHTH", "HTHTHH", "HTTHHH", "THHHH", "THHHTH", "THHTHH", "THTHHH", "TTHHHH"} *)
This is the final list that satisfies all the questioner's requests. Note that the "accepted" solution does not satisfy the request. For instance, the accepted solution does not include {HHHH} whereas the questioner explicitly demanded that this string be included.