From Longman Dictionary of Contemporary Englishpush on phrasal verb1 CONTINUE/NOT STOPto continue travelling somewhere, especially after you have had a rest We decided to push on a little further.2 CONTINUE/NOT STOPto continue doing an activity with Nixon pushed on with the weapons development program. → push→ See Verb table
Examples from the Corpus
push with• Even with disagreement growing, they decided to push on with the negotiations.• Hua pushes on with these and other whispered aids. 