Which is correct and why?
“I’ll have him call you.”
OR
“I’ll have him to call you.”
“They would have him do the job.”
OR
“They would have him to do the job.”
The first is correct in each case. Have (in this sense) is one of those words that can take an infinitive with a missing “sign of the infinitive” (to). It’s like the understood second person subject in imperative phrases. [You] do the job. Even though the meaning is technically the same with or without the you, it makes more sense if you don’t put it in.
I agree with Achernar; this is an error I often hear ESL speakers make, probably because the infinitive in many other languages does not have a “sign of the inifinitive” and they translate their language’s one-word infinitive to “to call,” etc.