How can I repair a damaged UF cable
Your UF cable can be easily spliced to either add in a new piece of cable or to rejoin a break in an existing cable. The splice kits are typically specified for copper wire only and consist of some set screw cylinders that join the wire ends and then some heat shrink tubing that has a very sticky adhesive inside. The splices can be buried and it may be preferable to do so from an appearance standpoint so just new wire shows above the ground level. 

One product I am aware of is from 3M company and is called their "UF Splice Kit" as shown below.



Another very similar product from Tyco is available at Home Depot stores. 

Do read the customer comments on the Home Depot page. One thing to learn from there is that a torch is not right tool to use for shrinking the tubing. Instead I recommend using an electrically heated hot air gun. (Although do use care using an electrical appliance outside, particularly of you are working in a wet hole in the ground). Another point of note is that successful sealing of the UF cable with these splice kits is to clean the wires very well before applying the tubing.