Straightened Curly Perm

A: I would advise against it. Generally speaking, the chemical processes in "perms" and "relaxers" are not cross-compatible. In specific use, you should never use a "perm" on hair that has been treated with a hydroxide relaxer.
I cannot in good conscience give you the go-ahead to use a relaxer on your hair after using a home perming kit. There is too much that could go wrong in the best of circumstances. However, your hair has been permed in the salon, retreated with a home perming solution, and is assuredly much more porous than you may think.
Because of this the timing and processing of the relaxer could easily be misjudged (to tragic effects). If you are still unhappy with the amount of curl remaining in the hair, I recommend you return to your salon and ask for their assistance in getting rid of the remaining curl.
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See also:
Hair horrors
Perms and perming hair
Hair straightening and thermal reconditioning
Hydroxide relaxer and hair straightening kit