Akademik

С-266
СЛАБ(ЫЙ) НА СЛЕЗУ (НА СЛЁЗЫ) coll AdjP subj-compl with copula (subj: human or detached modif (variants with long-form Adj only)) one cries often and usu. without any real reason
X слаб на слезу = X is easily moved to tears
tears come to X too easily it doesn't take much to make Xcry.
Он высморкался, рукавом чекменька раздавил щекотавшую щёку слезу, подумал: «Старею, видно. Слабый на слезу стал...» (Шолохов 2). Не blew his nose, smeared away a tear that was tickling his cheek with the sleeve of his coat, and thought to himself, "I must be getting old The tears come too easily. " (2a).

Большой русско-английский фразеологический словарь. — М.: ACT-ПРЕСС КНИГА. . 2004.