My 2003 Honda VTX 1300 S backfires through the carb what could be the cause and fix for this

replace the air filter, and when the covers off tap the bottom of the carbs with a hammer and piece of wood or plastic.

make sure there isnt any thing in the air box or exhaust little critters like to nest in them

You should change spark plugs. If you you have after market pipes you need to have it rejeted or the problem will continue.

great news you can only win with this. i bet if you pull a spark plug and look at it, it will be white or close to it. if so your motor is lean, it's not get the right amount of gas. when you put pipes on a bike it lets more air flow and you need to jet your carb. you local bike shop can help you get a jet kit and handle the install if needed. when done right you get more performance out of it. while you are look into jets get a high flow air filter or look into a hyper charger they come a jet kit, k&n air filter and the air scoop price isn't bad rsports.com/wps/Honda_OEM/HondaMC.asp?Type=13&A=474&B=25

Check the wiring to the spark plugs and replace if necessary.If that does not solve then go one step further backwards towards cdi etc...

Sounds like getting hot is the key here - try a better shop. A competent shop should be able to find this quickly. Sounds like a carb issue to me.

you have to get it re- jetted. I've never heard of anyone around here doing it themselves. its just relatively cheaper to get someone else to do it than to have to do it yourself. I've taken the carbs off my bike and it was a pain.

If you have used Dyno jet kit I have had a kit come through with smaller main jets than the stock ones causing it to run way to lean.

The pick-up coil next to the stator could be bad as well.