What chores does he enjoy doing, whether he is effective or not? At 4yo, it really is more about the effort, not the results. It also needs to be the right length of time. This is going to look different for every kid. They also like company. Chores at this age (and, why not? for as long as necessary) should come as a family activity. When everyone gets in on the action, being *out* of the action is less fun.
I have had success with expecting very little things: the girls put their day's clothes in the dirty laundry. It's a tiny, tiny chore, but it is incredibly helpful.
Now we have a toy chore chart (Melissa & Doug). They decide what chores they want to do during the day from a list. At the beginning of the day, I tell them what I need to do, and if there is anything they want to help with. My oldest (8.5yo) loves ironing and dusting. I do not need ironing done, but why not? When I was little, I liked sweeping and dusting. Not so much ironing, but then I was expected to do all my clothes plus pillowcases. I still like pillowcases. DD irons the handkerchiefs, which I also do not need ironed, but again, why not? (With our system, the girls reward themselves--I have nothing to do with this chart, but that is just our house and the way we work and live.)
Just like dinner on the table, make sure some of the options are ones he would enjoy, even if it would be more helpful if he chose something else. The goal at this point is to engage him in a necessary family activity.
Also, make sure that you don't make chores sound like drudgery. You don't have to be Snow White and be all cheery, but don't grumble and gripe, either. Put on some energetic music and sing and dance your way around the room, scooping things up, vacuuming. Kids love dusting, as non-essential as that is. They love anything with bubbles and brushes. Don't expect the "chore" to be something that *really* needs to be done at this point. Polished the mirror yesterday? Again today? Why not?
Good luck. Remember that a 4yo is more likely to make chores take longer. It's OK to get irritated when they mess up your good work, like jumping on a bed full of neatly folded laundry. They will not be 4yo forever.
And I personally would ditch the rewards for cleaning. Your son is the poster child for the downside of rewards for chores! Kids can and will start helping out without them. You need to have faith and patience with that. But in the end, you might decide that help with chores is more important and that they might outgrow the need for rewards for every effort. It's all up to you and your priorities. Cash for chores? Why not, if it eventually gets you the help you need? I had teenage friends growing up that were paid each $20 for getting the house clean. And they cleaned all of it (this was back in the early '80's when $20 was a *huge* amount of money!)
Unfortunately, when a lot of kids start helping out willingly is about the same time they get whisked off to school, adding more to their day and in making chores a burden for them.
Another personal note: I do not tie chores to allowance in any way. Allowance is for learning about money here, period. I know my money-motivated daughter would do more to help with that system, but I tell her, if I do not get paid for it, neither does she. Everybody feels better when the house is cleaner and more organized, that is our reward. She has other ways to earn money, as both dh and I are self-employed.