Could you please explain them?

I don't think 'for' is very natural there.

A :The cops called on me because I was acting suspiciously.
B : I had the cops call on me because I was acting suspiciously.

cf:

The cops arrested me for acting suspiciously.

messier42A : to call on me for ..B : to call me for ..

'To call on someone' mea

I think that "I had the cops called on me" can be interpreted as meaning that someone else called (telephoned) the cops and complained about you, or told them about some offence that you had committed. I wouldn't quite class this as standard English, but I think it might be used colloquially.

GPY"I had the cops called on me" can be interpreted as meaning that someone else called (telephoned) the cops and complained about you,

Ah, yes!

Thank you! But is there " have + objective + past present tense" ?
What's the meaning of " call on someone for something"?

messier42But is there " have + objective + past present tense" ?

What do you mean by that structure? Please give an example!

messier42What's the meaning of " call on someone for something"?

1. ask someone to do something: The board called on me for a rousing speech at the AGM.
2. visit someone for some purpose

messier42But is there " have + objective + past present tense" ?

I think you mistyped something here ("past present tense").

messier42What's the meaning of " call on someone for something"?

It depends on context. In your sentence, "called on me for acting suspiciously" is understood to mean something like "called on me

Like to have the copes called on me ..

messier42to have the cops called on me

This means that someone else calls the police and asks them to arrest you.

Is the usage the same as " to have my hair cut"? I don't really get why it has " have + objective + past present tense"

to have the police called
to have one's hair cut
to have one's house burnt down
to have a phone installed
to have the trip spoiled

There seems to me to be two meanings: one is the coercive (to arrange that someone else does the act) and the experiential (to experience the act). Context and common sense would indicate which is meant.