2004 Hyundai Santa Fe Sport- Bogs down

My vehicle has an interesting fuel system problem. First problem started years ago when I found out that you had to give the car a good bit extra gas while cranking over for awhile before starting and had to keep gassing until the vehicle stabilized the rpms itself. This of course eventually resulted in another new starter but that’s a tangent. This problem only occurred when putting gas in the car. Later I noticed that the engine would stall when you take the gas cap off. 3 mechanics had no idea and did not want to even look at the vehicle. Next problem was very infrequent where when turning the car off, it kept running. Last few months the vehicle bogs down a little every now and then after starting and last week it had one more indicator when for two days I noticed the bogging down in sync with the piston firing. My old car had a similar problem with the gas where the tank vent hose needed replaced but this doesn’t seem to be the same problem. So far the suggestions have been anywhere from a new fuel pump to a new gas cap to a new washer at one point in the fuel system. I don’t want the vehicle on blocks for a long time to diagnose so I’m hoping you guys can provide the miracle answer.

I would think you should be getting a check engine light. If so have the codes read and post them here. The actual code number, format P0123.