Is it ever grammatically correct to use the word "on" after the word "continue"?
"After this break, we will continue on with the broadcast."
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Though, to many, it is idiomatically redundant, it is not incorrect.
Here is a corroborating source: https://www.dailywritingtips.com/continue-and-continue-on/
It is certainly OK. There are many other examples of this kind of construction (see below). The word "on" is an adverbial particle and in this case and others remains fixed; it cannot move. For example it wouldn't be OK to say "After this break we will continue with the broadcast on." In many cases, the particle can move, as in "Please, take off your shoes," or "Please, take your shoes off."
Here are two examples similar to yours:
I hope she moves on with her life We need to continue on with the mission. After he caught his breath, he continued on with the race.
Your question asks about "continue" and whether or not it can be paired with "on." This is a matter of taste and a person's ideas about the verb continue. In my opinion "on" can be. This page had numerous examples which seemed fine to me. Can you leave out "on?" Sure. Adverbial particles give extra information about the verb. For example "up" indicates a completion of something: "finish up," "zip up," mess up," and "pack up." "On" as a particle indicates many things. In your example, it indicates the furtherance of an activity and helps the verb "continue" which means to keep doing something.
If you'd like to learn more, this is an article to explore.