1. You have to have your ticket on until you leave this club.
("on" is modifying "have" as in "have your ticket")
2. They wanted me on to remain as an assistant in the laboratory.
("on" is modifying "wanted")
3. He used to tell me on about the story until I went to sleep.
("on" is modifying "tell")
4. He said on about him until you called me on the phone.
("on" is modifying "said")
None of the four sentences you wrote is natural English. #1 is just about possible with 'you' after 'one'. #2 is possible if you delete 'on'.
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None of the four sentences you wrote is natural English.
#1 is just about possible with 'you' after 'one'.
#2 is possible if you delete 'on'.
#3 and #4 need to be recast completely.
These are natural (casual) English.
1. You have to keep / have your ticket on you until you leave this club.
2. They wanted me to stay on on to remain as an assistant in the laboratory.
3. He used to tell me on about the story
The following are okay:
1. You have to have your ticket on you until you leave this club. ("On you" is a prepositional phrase with an adverbial function, modifying "have.")
2. They wanted me to remain on as an assistant in the laboratory. ("On" is an adverb modifying "remain.")
3. He used to go on about the story until I went to sleep. ("On" is an adverb modifying
fire1By adding "on", can I express the continuing action of a subject?
Sometimes, but certainly not always. You need to learn the phrasal verbs and idioms that use "on" to express continuation. You can't arbitrarily add "on" anywhere you like.
hold on to / hang on to : continue to hold
Can you hold on to my books while I tie my shoe?