Wiktionary
vb. To persist; persevere.
WordNet
Usage examples of "push on".
Jim decided to ignore it, however, and push on to the main point of his being there.
And, despite their size, they were so light that a push on one would put the entire stack in disarray.
The only course possible to me, since I meant to push on into Egypt, was to leave the boy and girl in the hands of the kind Sisters of Charity, who, happily, are to be found everywhere.
It had to be erect at this time to push on the walls of the flexible cylinder in which it fitted.
Frank was eager to push on toward the oasis and spy out the ground for himself, but Yaqui did not favor the idea.
I had private reasons for wishing to push on as quickly as possible.
The road gave him new life, and briefly he speeded up, but it was only surface charge, and within the next hour he twice fell to the snow, to struggle up tight-jawed and push on.
On the contrary, he was obliged to push on, in order to make up for lost time.
I have never been able to do a strong mind push on anyone, but I tried now.
My drive now was to rejoin my wolf, see that he was well, and then push on in pursuit of the Prince.
Or should I push on fast, hopefully catch the other party and make inquiry there?
You learn how hard to push on things -- hard enough so the pins will stay up, but not so hard that they won't go up in the first place.
There is no indication so far that he has any wish to reside here (the Unknown) or to push on into it because he has a Burtonian curiosity.