Meaning:
‘Hang on’ simply means ‘wait’ but it usually means to wait for a short time.

Conversation 1
a: “Are you ready to go yet?
b: “No, hang on while I find my keys.
a: “Ok, but hurry up!”

Conversation 2
Often we use it with a phrase about time to show how long we are going to be.
a: “Are you coming?”
b: “Hang on a minute, I’m not ready.

Conversation 3
We also use it with the words “a tick” (meaning a second, like the ‘tick-tock’ of a clock) “a sec” (second) or “a mo” (moment) for something that will happen in a very short time.
a: “Have you finished that work yet?”
b: “No, hang on a mo I’m just finishing now.
a: “Good.”

Conversation 4
Because “Hang on” is a phrasal verb we can use it in different tenses if we change the tense of ‘hang’.
a: “Are you coming?”
b: “No, I’m just hanging on (waiting) for the washing machine to finish.”
a: “How long will it be?
b: “Not long, just a tick.”
(present progressive)

a: “Did you see John?”
b: “No, I hung on (waited) for as long as I could, but I had to catch a bus.
(past tense)

(Callan book Stage 7, 608p)