My trolley fell over yesterday and the plug into the battery came out, there is visible damage to the plug-in on the wire from the battery.

It will still plugin also does arc as I plug it in. The trolley will turn on but will not engage when I press the button for it to move.

Do you think this is due to the damage to the wire? I am going to replace it, was just hoping for some guidance as to what other issue it could potentially be?

Problem and faults with the battery connectors

Concentrating on only the battery and its connection. It is most likely damaged the battery lead 3 pin round connector, probably fractured it and the metal contact sleeves within this black round connector are touching and shorting. I would definitely say a new battery cable will be needed. Now as for the female socket on the battery holder.

Connector sleeves inside the battery cable

How to Fix a golf trolley battery connector

Without seeing it I would check to make sure the pins coming out of the round socket are have not been damaged in any way.

1. Check if there is any visible bend in these pins. 
2. Check if the golf trolley battery cable round connector is fitting easily, where it plugs in or unplugs with no problem.

These will not guarantee there is no damage to these connection sockets, but a basic check should be 80% sure it is working.

The battery cable and connector are in stock, also the round socket that fits onto the battery tray is available in the online store.