K (ngn/k), 23 bytes

{x_/3#-1+*(2'x)?,"^H"}/

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Feels a bit hacky; removes one instance of ?^H on each invocation, and is run until convergence (i.e. no more ^H's are present).

• {...}/ run function on (implicit) input until it converges
• (2'x) take 2-length sliding windows of the input
• -1+*(...)?,"^H" get the index of the character immediately prior to the first ^H in x
• x_/3# drop the value occurring at that index three times

K (ngn/k), 23 16 bytes

-7 bytes by rereading question (specifically, "^ will only appear as ^H")

{x_/3#-1+x?"^"}/

Try it online!

Removes one instance of ?^H on each invocation, and is run until convergence (i.e. no more ^'s are present).

• {...}/ run function on (implicit) input until it converges
• -1+x?"^" get the index of the character immediately prior to the first ^ in x
• x_/3# drop the value occurring at that index three times