The post-credits scene for "Inside Out 2" explains what that "deep, dark secret" was from the first act of the movie where the core emotions are in a vault with Riley’s secrets.
In the vault, the core emotions are with Riley’s Deep, Dark Secret who looks like a black and white version of Hulk wearing a hoodie, but we don’t know what the secret actually is.
By the time the core emotions escape the vault and go to save Riley from Anxiety, we're left without any clarification. However, the film's post-credits scene solves the mystery.
Joy returns to the vault where Riley's deep, dark secret resides, to learn why it's been locked up securely. It turns out, it’s what any kid would want to keep from their parents.
When Joy asks the secret itself to explain, he admits, "We burnt the hole in the rug!" It’s a silly secret only a kid could worry about that also references "Back to the Future."