A wall anchor is a convenient device that allows you to hang larger and heavier objects on walls made of drywall or sheetrock that would not normally support the weight. They are generally quite easy to install and are very durable, but in the case that a wall anchor becomes cracked or damaged, it is better to replace it entirely than attempt to repair it. You could use Super Glue to repair a plastic wall mount or a metal epoxy to fix a metal one, but wall anchors are so inexpensive, it makes more sense to replace them entirely. You don't want to trust your wall hangings to a cheap part that has had its integrity compromised.
Step 1 - Remove the Old Anchor
If the existing anchor did not pull out entirely, you will need to first remove it and patch the hole up before you can install the new one. Work the old anchor out gently. Use a pair of pliers, or insert a screw into the mount and use that to pull it out. Once it's removed, you can patch up the hole.
Step 2 - Patch the Hole
Before you can install the new wall anchor, you need to patch up the hole where the old one was. If you intend to install the new anchor in the same location, you will want to do a more thorough job than if you are just patching it up for aesthetic reasons. Use the putty knife to scoop out enough filling putty to fill the hole. Scrape it over, flush against the wall, pushing the putty into the wall. Do this until the hole is filled to your satisfaction, and then let it dry.
Step 3 - Make a New Hole
Once the putty you placed in the old hole has dried, it's time to place the new one. You will want to be somewhat more careful if you are placing the anchor in the same spot as before, since the wall filler can crumble if subjected to too much force. If you are using plastic wall anchors, you want to make a hole that is about as big as the anchor itself. If using a threaded anchor, you can get away with only making a starting hole.
Step 4 - Install the New Anchor
Once you have placed the new hole, you can install the new anchor. If using a plastic wall anchor, simply tap it in gently with the hammer until it is flush against the wall. If using a threaded anchor, screw it in using the starter hole to get it going. In either case, one the wall anchor is completely installed, screw in the screws that were included. Your wall anchor is ready for use.