Enhance performance of update query
It depends on what you mean by improve performance.  You can't make that query faster, it needs to scan the table.  But you can keep it from using too many resources and blocking other processes by chunking it, which is essentially iterating through batches of  rows to perform multiple smaller updates.  

There must be some option to just write it to the target table instead of updating in place.  

You could also forego the scrub altogether and use a non-persisted computed column in the target database to reformat the phone number if you don't need to index it.