To continue a journey or task despite tiredness or obstacles.
"The hikers were exhausted, but they decided to push on and reach the summit before dark."
We shall go on to the end... we shall never surrender.
— Winston Churchill, speech to the House of Commons, June 4, 1940 (paraphrase of the 'push on' spirit; direct use of the PV in the speech is not confirmed — treat as illustrative)