What would someone mean if he said ' I had better get on eith some work' and it is right to say ' I had better go get on with somework'
EX: I had better get on with some work.
"get on" means, continue on; do.
EX: I had better go (and) get on with some work. <correct>
Two events: (1) Go, (2) Continue on; Do.
Note, and is often omitted.