That's the starter RELAY; the solenoid is ON the starter.
The problem is that nasty (+) battery clamp. Go to NAPA & buy this:
When you remove the clamp bolt, pull the wire & its ring terminal out of the clamp. Use a wire brush (preferrably brass, but steel if necessary) to clean the ring terminal thoroughly. DO NOT apply grease to it. Attach it securely to the new clamp. DO NOT apply grease to it. Clean the battery post thoroughly & spray on "Battery Terminal Cleaner & Leak Detector", then install the new clamp w/wire.
You won't use the ring terminals included with the new clamp now, but save them for other projects.