Gas does not flow freely on the right side

When I go to start my Norton 750, I noticed that on the right side it takes more pushes on those buttons to get the fuel to come out when the left seems to flow out easily. What do I need to do to clean this out so it flows easier?

Well it could be your petcocks test by removing the lines and doing a one minute of flow test. Fuel from good petcocks should proceed without ripples or splashing. Have a twelve OZ container on the ready and a good way to measure the results.
Or it could be your ticklers if the roll pin is shoved too far into the button than a pressing will not push the floats down as far as the other by comparison.
Or you may have some vapor lock restricting the flow.
Or you could have Banjo spigots that are drilled for a lower CCed bike.
Or your floats may not be set as a pair in terms of the level of fuel they maintain.
With 40 years to change the parts since it was new all things have been tried.

Sammy, not only have you purchased a Norton, you now have a hobby.
In order for this or any forum to be more helpful, you need answer a question. No matter what you say, we can help whether a complete novice or an expert. There are many answers to the same question as you see by the previous responce. Our ability to help you will depend on your ability to understand what we are telling you. So,my question to you is, what is your mechanical level and are you experienced with Norton or British motorcycles?

Could this have anything to do with an occasional backfire from the right side?

Pretty mechanical... No experience with British bikes.

Yes, that is possible. If you are starving on the right side the lean condition will cause backfire. Maybe one of the first thing you do is to get close and personal with your Amals. Check for flow to the carbs. Then you may want to dismantle, clear the jets (including the pilot with a #78) drill and get the needle/seat and floats proper.
Get a manual or 2 and a workshop manual ASAP.
Search on the Amal site for complete information. Much info out there.

When you say it takes more pushes on the buttons.... Are you pumping them up and down? You should be just holding the button down (to sink the float) until the gas comes out. I've seen a quite a few people doing this. It's not like a string trimmer or chain saw where you do need to pump the fuel in.

Welcome to the quirky world of Nortons where you will quickly learn to be a mechanic.

The petcocks could be partially blocked.
I just replaced this week two I purchased back in 2004.
Already blocked from these fuels today. :evil:
The rubber o-rings swelled so bad flow was at about 1/2 speed.
If yours are orig. I would start with these. Strainers are probably shot.
Let me regress, claen fuel tank 1st before investing any $. Old dirt on new parts will not correct the problem.
Treat it like the military taught me start at the top and work down.
38 y/o petcocks could use a renewing. Get the BAP's, cheapest from Waldridge Motors in Canada.
Replace that crusty yellow fuel line w/ an OEM.
Replace those strainers in the bowls.
Clean all jets and orifices in carbs as already explained.
New air cleaner will make a difference.
Make sure the main jet holder has not vibrated loose in the carb body causing a rich cond.
That is how I took delivery of my bike rich on the left side, holder loosed up.
Just tighten not overtighten as carb body is a cheap metal.
If you are getting good fuel set the idle screws and start.
Air screw thats right air screw not fuel screws should be set at 1 1/2 turns out from fully closed to get her running.
Did you buy the Clymer Manual I told you about yet?
It'll give an excellent step by step on how to dial them in.
Also a couple good sites if you google Amals on how to dial them in.
You gotta love it or it's no fun.
Welcome to the unapproachable world of Nortons.
Marshal