It doesn't happen with acceleration from a stop but when one is just riding down the road. Tom says this car has the automatic CVT transmission. We've also done the disconnect the battery to reset thing, which helped briefly.
Thank you for putting the time into this! It's a help to know what you found.
That makes your problem a little different than the common problem, which only happens when accelerating from speeds below 5mph. Hard to say exactly where it would be, but I would first check the air and fuel systems and sensors. Especially the sensors, since the battery reset helped for a while. It's very likely one of them is near failure or has an intermittent connection problem. This wouldn't necessarily throw a 'check engine' code, but daily scans might reveal a 'pending' trouble code. Also check the 02 sensors, they're used to regulate the fuel/air mixture, and their wires tend to get exposed to a lot of heat. Corrosion on electrical plugs is also a very frustrating nuisance in the NorthEast.
If there's nothing there, look very closely at the whole air intake assembly. Very small leaks around the air intake can sometimes cause low or negative airflow in certain conditions (basically, the right cross-draft can suck the air out of the engine), causing intermittent loss of power and hesitant acceleration.
Just thought I'd mention... if you don't have an OBDII scanner (code reader) at home, they sell bluetooth enabled scanners for around $10 that work with a free app on your phone, and will do everything (or more) than your local mechanic's $300 scan tool. They can be super helpful for seeing what's going with the engine and powertrain while the hesitation is actually happening.
My husband had a little thing like that, no idea what version. Tom says it's still in the car, if I ever get it back from the mechanic's...
I've passed this conversation onto Tom...