I am assuming that the engine is bogging down and maybe even stalling out when you try to accelerate quickly....
Your mechanic needs to check the dashpot valve ...#41 It is a one way valve.
http://www.boats.net/parts/search/Honda/Outboard Engine/2001/BF50A1 LRA/CARBURETOR/parts.html
He also needs to check #1 Diaphragm to be sure that is is operating and properly adjusted.
The diaphragm and parts 4 and 6 and the dashpot valve (above) work together when accelerating and quickly decelerating to help keep the engine from stalling.
http://www.boats.net/parts/search/Honda/Outboard Engine/2001/BF50A1 LRA/THROTTLE ROD/parts.html
A couple of other things could cause your problem.....
- Oil level too high
- Timing belt off a tooth
- Valve clearances out of adjustment.
- Compression issues
- Motor too small for the boat
Since you have already changed out the carburetors, be sure that they are the correct carburetors for that engine. Therefore, the idle mixture screws should be adjusted already. Also, check to be sure that there are no restrictions in the fuel line from the tank to the engine.
Oh yes! The propeller may be too big of a pitch and the engine may not be able to turn it quick enough in acceleration...and/or there may be a lot of growth on the bottom of the boat, causing the motor to work harder. Hopefully, you are getting somewhere between 5500 and 6000 rpm at full throttle.
I have to assume, that if the engine runs ok at high speeds, once you get there, that there are probably no air leaks in the fuel system.
I am also assuming that you have had this motor for a while and it worked ok at one time. Be sure to start out with motor trimmed all the way down....then trim up as you get on plane. I don't mean to insult your intelligence, however, many people do not know that....and I do not know your boating experience.
Hope that helps some.
Mike
