Word for eyes "pleading for approval"

pleading OED adj

that pleads; esp. entreating, beseeching, imploring.

As in:

• 1901 ‘H. McHugh’ John Henry I hate to have a girl plant her pleading peepers on me.

• 1968 A. K. Armah Beautyful Ones are not yet Born The boatman's eyes were no longer the diffuse, vaguely pleading eyes of the hireling the man had seen the first time.

Pleading eyes is synonymous with that begging look, forlorn puppy look.

https://www.collinsdictionary.com/us/dictionary/english/imploring

imploring (ɪmplɔrɪŋ ) ADJECTIVE [ADJ n]

An imploring look, cry, or letter shows that you want someone to do something very much and are afraid they may not do it.

Frank looked at Jim with imploring eyes.

imploringly ADVERB [ADV after v] Michael looked at him imploringly, eyes brimming with tears.