The top rail on the chain link fence is damage and must be replaced.Can I cut out the damage part with reciprocating saw and replace the bad section with a top rail sleeve?

There's different kinds. Is your rail going through loops that are caps on each post? Do you have a picture?

There is a repair part at HD. 1 3/8 and 1 5/8 top rail. It fits over the smaller and inside the larger. I have used it before. Cut the top rail and use them or see if you can release the fabric at one end and replace the whole rail.