'have him watch' is correct and usable in written English.
You can use it to refer to someone viewing something. For example: "I'm going to show him the photos, so have him watch carefully."
Having him watch me was like being tickled; I felt happier.
Johnson said that Moawad had him watch highlights of himself scoring goals.
Plans include having him watch a math class concocted for him on a day of no-class final exams.
I've had him watching them as a completely outside pair of eyes.
"We've had him watched after contacts brought him to our attention.
"And we've had him watched on a number of occasions for Bolton's development side.
"I had him watched two or three times last year, I've had him watched two or three times this year and it was just a case of asking Argyle if we could do something about it," he concluded.
"We have had him watched and he will add something different to what we already have available to us".
"Liverpool have had him watched and like him, and they have asked us if they can have a closer look at him," Hargreaves told the club website. .
