Not sure about English but in French, correspondance while obviously closely related to “correspondence” is mostly reserved for written communication so would rule out phone calls. Your friend rightly used contacter which covers all the methods you cite.

Je vous prie de faire preuve de patience en me contactant.

This means the people contacting you are kindly asked to be patient while doing it. I wouldn’t write it that way because it is not while contacting you that they need to be patient but later, when waiting for a reply.

Your patience is asked to answer the correspondence.

This sentence doesn’t translate the previous one and seems odd to me. It looks like the requester is also the one who answer.

Votre patience est prié de répondre à la correspondance.

This sentence makes no sense in French. The patience is impersonated and asked to answer itself to the correspondence.

Here is a couple of ways to express what you want to say:

Si vous me contactez, je vous remercie de faire preuve de patience en attendant ma réponse.

or the more formal

Si vous essayez de prendre contact avec moi, je vous serais reconnaissant de bien vouloir faire preuve de patience en attendant ma réponse.