Chrome plating is the process of bonding a layer of chromium onto other objects to give them a chrome finish. For faucet fixtures, chrome provides several benefits -- it has a sheen that works well in kitchens and bathrooms, it is relatively inexpensive to manufacture, and it is highly resistant to corrosion. However, the chrome plating can become scratched or damaged with frequent use. Serious damage requires a full replacement of the fixture.
Chrome plating is not designed to be patched. If you have a small scratch on the chrome surface, then purchase some chrome polish and a scrubbing pad. Thoroughly clean the chrome, apply the polish, and buff the faucet. If the combination of polish and elbow grease does not remove the scratch, or if you have a more serious gouge, you may not be able to fix it through repairs.
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Chrome is not a paint that can be peeled away and re-applied. While there are chrome paints designed to give a chrome-like finish to certain objects -- especially cars -- these are not actually chrome and give poor results when used on real chrome objects like faucets. Patching materials, like putties, are primarily designed for other uses, such as grouting or plumbing repair.
If it's not the chrome surface you're concerned with, but rather the faucet itself, you can use patching materials on the general faucet system. Packing grease can help protect and seal faucet components if you need to replace any chrome parts. Silicone caulks can help protect the base of a faucet from spills and water leak issues.
Replacing the Faucet
If a damaged chrome surface is bothering you, or has left part of the faucet metal beneath exposed to corrosion, you can replace the entire faucet, or at least the piece that has been damaged. Find the brand name and model of your faucet, and explore manufacturer products for any components you can buy individually. Compare costs for replacing the faucet entirely with a new chrome-plated model without the damage.