You can say:
This is the phone (that) I most wanted (to have) my mother to buy me.
Both 'that' and 'to have' are optional and don't change the meeting. English leaves out the word 'that' when introducing subordinate phrases, whereas Spanish generally requires it. In informal speech, speakers often insert 'to have'. It is not needed. Other options are:
This is the phone (that) I most wanted (to get) my mother to buy me.
This is the phone (that) I most wanted (to make) my mother to buy me.
This is the phone (that) I most wanted (to convince) my mother to buy me.
This is the phone (that) I most wanted (to implore) my mother to buy me.
These optional words 'get / make / convince / implore ...' tell us something about the desires or mental state of the person saying the sentence.