If it is rotten you are better off replacing it. More work, I know, but it's a permanent fix.

if the post is ok, then you need to dig out several inches around it as much as possible. dig under the concrete walk from teh side, then pour some fresh concrete in there. this will brace it up some.

honestly the problem look to me tha tyou have a 4x4 for a corner/end post. you should have a 6x6 there for maximum strength.

You can change the post with a recommended 4 x 4 pressure treated post to avoid the same problem in the future. Secondly what you can do is actual buy a metal bracket available at home depot or lowes that you use to jam into the concrete with a sledge hammer, then screw into your post to the metal bracket. Hope this was helpful

Here is a link that might be useful: choosepatriot.com