In aid climbing this is still sending since hanging on protection is part of the game (as @StrongBad points out, hanging on the rope is a little different than hanging on protection.)
In sport climbing , however, there is no specific word or phrase .
I think the most common ways to describe it are: Projecting
As in "I'm projecting Era Vella 9a.
" This doesn't capture the fact that you've done all the moves, but most sport climbers are focused on the clean send, and so a large part of projecting involves doing all the moves in sequence with the occasional hang.
Worked through all the moves
As in "I worked through all the moves on Predator 5.13b." This could mean that you rappelled in to try each move, but in some groups this will be understood as ground-up climbing. ...
but not clean.
As in "I climbed my first 5.8 today, but not clean.
I had to take once after the crux.
" This will usually be followed by rude people reminding you that you didn't "actually send", and nice people reminding you that you should climb however you want as long as its safe and fun.
Edit: I changed the example here from "sent" to "climbed" because after thinking more about Adam's comment, I think there is a whole connotation involved with "sending" that some climbers may not like appropriated.
I think it's important to realize, though, that not officially "sending" is no less impressive or cool than sending.
My experience in sport climbing circles is that no matter how hard you send or how many times you hang, other climbers will be stoked for you if you're trying hard and enjoying yourself.
outdoors.stackexchange.com/users/18067/jhch
Without falling, wouldn’t it be hangdog, see https://en.m.wikipedia.org/wiki/Glossary_of_climbing_terms#hangdog
With falling, I also would go for projecting.
(Speaking of Sports climbing!)
outdoors.stackexchange.com/users/11707/canIchangethis
That is a free climb (the outcome).
The term you may use is All Free (AF).
It is much more established in German (Alles Frei, remember, that RP »Rot Punkt« also comes from German).
It (AF) makes more sense in traditional sandstone climbing in Elbsandstein, in multipitch (big wall) climbing and in alpine climbing though.
Check, for example, this book , I quote "The AF (All Free) style was developed in the Alps as the first recognized sport style of free climbing , where the climber uses only natural for advancement and and overcomes the route with his/her own strength."
outdoors.stackexchange.com/users/3055/Vladimir F
I would say "I've made all the moves, but not put them together yet".
Usually, when I'm projecting a route, and got into that situation, I think of it as "I've made the moves, but used 120% of my energy.
My challenge then is to make the moves more efficient; saving 2% here, 3% there, until I get it down to 99 or 100%: then all I have to do is make it perfectly :)
No slips, no moves out of place.
It works for me
outdoors.stackexchange.com/users/18362/Lawrence Sweeney
It depends on the number of rests taken In sport climbing, when you climb a route but hang on the rope to take a rest at one point during the ascent, it is called a one-hang .
If you rested twice during the ascent, it could likewise be called a two-hang .
If you rest on every bolt of the entire route, it is called climbing bolt-to-bolt .
outdoors.stackexchange.com/users/18567/Moormanly
I think the general progression I use is climbed, led, freed, flashed, and on sighted.
I might break freed into red point and pink point if the route or area makes if ambiguous.
I generally break led into a lot of terms like groveled, hang dogged, whimpered, depending on how ugly I was.
Terms like projecting to me
mean you are working towards freeing the route, but if I led a route with a single fall and have no plans to go back, I would describe that as projecting.
outdoors.stackexchange.com/users/5836/StrongBad
If a climber completes a route without using their protection that is called sending.  What is the term if one completes the route but had to depend on the rope either for a rest or for a fall?  Let's say using the rope lowers the climber enough that they still have to complete all moves. In other words all moves have been done properly, but the ascent wasn't clean.
What is the term for completing a climbing route uncleanly?
outdoors.stackexchange.com
<climbing><terminology><free-climbing>
outdoors.stackexchange.com/questions/22744
outdoors.stackexchange.com/users/18347/Adam
The term you may use is All Free (AF). It depends on the number of rests taken.  In sport climbing, when you climb a route but hang on the rope to take a rest at one point during the ascent, it is called a  one-hang . If you rested twice during the ascent, it could likewise be called a  two-hang .  If you rest on every bolt of the entire route, it is called climbing  bolt-to-bolt . Without falling, it would be hangdog and with falling, it would be projecting. Projecting, hangdog, one-hang/two-hang and bolt-to-bolt are all options.
Projecting, hangdog, one-hang/two-hang and bolt-to-bolt are all options.