Only just seen this, so apologies for tardy reply.
With a healthy lamb you’d probably just leave it but keep a weather eye on it (apologies for terrible pun, I simply couldn’t resist
). But as this boy’s a bit of a flake, I’d probably wash it with salty water, let it air dry, then if I had some, give it a spray with the blue antibiotic spray. If no blue spray (and no nearby farmer who’d let you have a squirt of his/hers), then iodine and repeat the salty water/iodine daily until it’s clean.
Personally I wouldn’t use systemic antibiotics (injection) unless there is reason to believe the infection is spread inside. As he’s a sickly one, you might think it better to be safe than sorry on this and I don’t think any of us would hold that against you.