Possessive indirect object pronouns?

Possessive indirect object pronouns?
 

I was working on an online exercise today where the object was to rewrite sentences and replace as much as possible with direct and indirect object pronouns. I was doing okay until I came across this sentence:

I tried interpreting this as "they are going to cancel the cable to me" or "they are going to cancel the cable for me" or something like that, but it didn't seem correct. I typed it into Google Translate and it said "They are going to cancel my cable."

Is Google correct? Is this a common construction? Is "ellos van a cancelar mi cable" also correct? Am I thinking about this totally wrong?

The translation is correct.
In Spanish, it's common to say that something will be done 'to you' or 'for you'. The possessive pronoun usage that you asked about is a valid sentence, but it wouldn't be the preferred form.
If you think about the purpose of a indirect object, it is the indirect recipient of an action performed on something else (the direct object). The direct object answers the question "what?" What is being canceled? Cable. The indirect object answers the question "for whom?" For whom is the cable being cancelled? Me.

Gracias! That really helps.

I think this is an example of the "ethical dative", which is the use of an indirect object to mean something like "in a way which affects X's interests". Unfortunately Wikipedia doesn't have an article about it, but you might be able to find something useful elsewhere.

I have other idea about the build of the sentence wrote in the first post.

Ellos me van a cancelar el cable.

They are going to cancelling me the cable.

They are going to cancel my cable

The second ones is like the first one, but they are written of the different form.

Is it right?