Explaining that one of
its many difficulties had been the recent growing injury list among starting players, the minor league baseball team said it would start a month-long search for better physical trainers and physicians.
(A) its many difficulties had been the recent -> Its refers to XYZ team, It is okay. Further, we are in past tense "said" gives a hint, so "had been" is correct form of tense. Let's keep it.
(B) its many difficulties has been the recently -> Tense agreement issue. We need past tense.
(C) its many difficulties is the recently -> Same as B.
(D) their many difficulties is the recent -> Pronoun error. "their" doesn't refer to "XYZ team".
(E) their many difficulties had been the recent-> Same as D.
So, I think A.