[Supperman]

" Hang on a minute- won't be long" (correct)
meaning we want them to wait for a short time until we have finished what we are doing, So in that context we are asking them to wait a minute or two for us.
"Hang on, I want to look in this shop." expecting a friend to wait a minute while they look in the certain shop. (correct)

If I hang on to continue this thread, I would become a Master of English!. (wrong)
If I persevere with this thread, I will make a good progress. (correct)

"Hang on in there" (correct)
" Hang on in there, don't give up, you can do it--" (correct)

"A lot of expensive demand notes have arrived me" (wrong)
"A lot of expensive demand notes have arrived for me" (awkward)
"I have received a lot of expensive demand notes". (correct)

Pete: What's happened?
Suzie: Bob has fallen off a cliff and is hanging on by his fingers!
Pete: Hang on, Bob! Help is on it's way!

"Hold on to the dog's collar, he might bite me."

Mum: Hey, hurry up and get off the computer! >[
Me: Hang on, hang on! It's just printing! >P

Sue: Should I put the fish in now?
Anna: Hang on, let me read the recipe.

Let's go back to Bob. The dog bit him and he's now at the hospital.
Bob: Doctor, it hurts! Please can I have some pain relief?
Doctor: Just hang on, Bob, we must stop the bleeding first.

'hang on studying' (wrong)
hang on ---ing (wrong)

"Bob couldn't hold onto the cliff, and he fell."
"The Joker couldn't hold onto the ladder and he fell."
"Bill didn't hold onto his dog, and it bit someone."

"Suzie's a nice girl but she just can't hold onto a job."
She quits or gets fired a lot.
"I can't seem to hold onto my money at the moment- as fast as I make it, I spend it!"
"Bob didn't hold onto his wife, and now she's left him for Bill."


"I can't hold on anymore" means "I am about to break" "I want this to end" "I've had enough of this situation."

Bob: I just can't hold on anymore; I nearly died falling off a cliff, a dog bit me, I lost my job and my wife has left me for another man. I've tried to stay cheerful, but it's just too much- I wish my life was over!