I recognised this premise from the 1990s cartoon episode Make a Wish and Attack of the Octobot.
This episode [Attack of the Octobot] and Make a Wish are based on a combination of two different storylines from The Amazing Spider-Man comics which are The Amazing Spider-Man #55 - 56 (December, 1967 - January, 1968) and The Amazing Spider-Man #248 (January, 1984).
In The Amazing Spider-Man #248, which is titled The Kid Who Collects Spider-Man!, Spider-Man visits a young boy named Timmy Harrison who idolizes him. At the end of the story Spider-Man reveals to Timmy his secret identity and you learn that Timmy is dying of leukemia and only has a few weeks left to live.
So the comic where he takes off his mask to a child is 'The Amazing Spider-Man #248', however it is only in the cartoon that the child is a girl.
Neither episode has a line about Peter Parker having a common face though, but somebody in chat found the panels for me:
Spider-Man: Now Tim, you know that's my greatest secret. If people knew who was under this mask. I'd never know a moment's peace. All of my friends would live in constant danger
Tim: I know... But I'd never tell another soul that I knew... Long as I lived.. Honest!
Spider-Man: I believe you Tim, but I can't --! I...
Spider-Man walks to the window, then returns to Tim's bed and removes his mask
Spider-Man: Tim... My name is Peter Parker. I took most of the photos from the articles you saved.
Tim: You mean all these years, 'ol man Jameson has been paying you to take pictures of yourself?
Tim: Boy that's great, I wish we could tell him!
Spider-Man: I know, I do to... But we can't! You mustn't even tell anyone I was here!
Tim: Don't worry Pete. I understand. It's our secret. Forever and ever I promise.
Spider-Man: Yeah. I guess that makes us best buddies then!
Tim: You bet Pete. Buddies to the end.
Spider-Man: Hey... It's really... getting late now! I have to get moving, and you have to get some sleep.
Tim: Okay g'night Pete!
Spider-Man: Take are of yourself, Tim.