The answer to this question is a qualified yes.  Attorney fees, which are unearned by virtue of the malpractice, or other fees which are spent to rectify the situation may be collected.  Here is a case from the Third Deparment on the issue. LEACH  Appellant, v.BAILLY et al., Respondents.

"Finally, turning to plaintiff’s arguments regarding the malpractice committed against him individually, we reject plaintiff’s assertion that he is entitled to fees incurred in subsequent litigation unattributable to that malpractice. We conclude, however, that he is entitled both to [*3]counsel fees paid to defendants and to counsel fees incurred upon the appeal of the adverse determination in the underlying action, which resulted from the admitted malpractice committed against plaintiff individually (see Rudolf v Shayne, Dachs, Stanisci, Corker & Sauer, 8 NY3d at 443-444; Harris v Bonacci, 65 NY2d 876 [1985], affg on op below 109 AD2d 1072, 1073-1074 [1985]). Unearned fees may be recovered in a malpractice action, and "plaintiff’s damages may include ‘litigation expenses incurred in an attempt to avoid, minimize, or reduce the damage caused by the attorney’s wrongful conduct’" (Rudolf v Shayne, Dachs, Stanisci, Corker & Sauer, 8 NY3d at 443, quoting DePinto v Rosenthal & Curry, 237 AD2d 482, 482 [1997]; see 3 Mallen and Smith, Legal Malpractice § 21:6 [2008]). The record establishes in particular, the parties’ stipulation that defendants committed malpractice in representing plaintiff that such fees and expenses total $15,342.43, and plaintiff is entitled to interest from the date that his damages were incurred, i.e., the date that the fees were paid (see Rudolf v Shayne, Dachs, Stanisci, Corker & Sauer, 8 NY3d at 442; Harris v Bonacci, 109 AD2d at 1074; CPLR 5001 [b]). "