meaning of Ihrem
I think the translation of "Ihrem" is your (singular).
But what about in the sentence written below ?
Hören Sie und sprechen Sie dann mit İhrem Namen - their or your ?
This is not true. "ihrem" (not capitalized) means "her" (3rd person singular female dative) as well as "their" (3rd person plural dative). Have a look at a table like https://study.com/academy/lesson/german-possessive-pronouns.html or simply believe a German native speaker.
The context you gave shows me how this is is probably to be interpreted. It follows an audio example and you are then given the order to listen to this first and then perhaps apply it to yourself, using your own name instead of that of the person in the audio. In this context the sentence makes some sense.