The Puppet

The Puppet is the secondary antagonist of FNaFb 1 and 2, and a minor antagonist in FNaFb 3. He is BB's right-hand man that tends to pursue the party when his music box isn't wound up.

In Five Nights at Fuckboy's 1 he appears in the Bathroom after defeating two cameras. You have to wind his music box every five minutes or he will come and attack. That is, if you wanna keep him neutral. He has 80.000 HP. When he is fought, it shows his jumpscare from Five Night's at Freddy's 2.

In Five Night's at Fuckboy's 2, the Puppet appears at the Prize Counter after defeating two cameras. Same as the first game, you must wind up the music box before the five minutes are up. The Puppet has about 50.000 HP. When he is fought, it shows his position when he exits the music box in Five Night's at Freddy's 2, with two very low opacity "clones" in his back, reffering as a small reflection.

In Five Night's at Fuckboy's 2, there is another puppet: "The Second Puppet." He is a a lot harder, and he appears once The Puppet has been defeated. He has only four minutes before he attacks. He is almost twice as strong and very defensive, and has 150.000 HP. When he is fought, it shows his jumpscare from Five Night's at Freddy's 2, but the background is darker and The Puppet is colored slightly differently.

In Five Night's at Fuckboy's 3, probably like you suspected, you battle Phantom Puppet. He appears once two phantom animatronics have been defeated. He has five minutes on his timer. He has 50.000 HP, and when he is fought, it shows his image in the "Animatronics" menu in Five Nights at Freddy's 3.

After you defeat Phantom Puppet, he reappears as Past Puppet, much to Springtrap's surprise. He is extremely hard, even harder than all the other Puppet's in the series, with 120.000 HP. The Past Puppet looks the same like the Second Puppet in fnafb 2. Once he is defeated, the 90's Endoskeleton Battle is unlocked and you can fight Phantom Tupac, who has 80000 HP.

Trivia

 * Initially, when wounding The Puppet from Fnafb 3, a slapping sound would be heard. This was deleted in the last update.
 * in Five Nights at Fuckboy's 3 there was another puppet called "Future Puppet" but got unused and got reused in act 0