Although it is possible to effect a repair to minor damage on gutters and downpipes, these should really be viewed as a temporary, rather than a long-term, measure.
Spray on repair
There is a spray asphalt available specifically for this job. It is suitable for small splits and cracks.
Clean the area to be repaired by removing dirt and debris. Wipe the area with white spirit to clean off grease and oil. Allow to evaporate and dry.
Mask off the area to be sprayed with old newspaper or card so that the asphalt does not mark sound areas.
Follow the manufacturers instructions. Shake the can and spray a coat of the asphalt from about 300mm. If the damage is significant, lay a mat of glass fibre over the asphalt before it dries and press down.
Apply a second coat once the first has dried.
Epoxy putty repair
A broken edge of guttering can be repaired using epoxy putty. This is a two pack material which can be moulded and sculpted to shape before it dries.
Again, mask off undamaged areas. Using a piece of card lubricated with light oil to prevent it sticking, make a temporary support to the outside following the shape of the gutter and fix in place with masking tape. Mix the two components as directed on the pack and shape it to match the original profile. Allow to set before removing the support which should peel away easily.