Sounds like oil ring isn't doing the job in at least one cylinder.

My recipe:

1. measure bores with a quality bore gage (size, runout top to bottom and for ovaling
2. Bore and hone if out of service limit
3. Hone if within service limit (if your machine shop doesn't know Hondas, warn them to take just a couple-three strokes on the hone with NEW stones before measuring. I watched a NASCAR engine builder who thought he knew better take too much out of a Honda block. He was stunned how fast the metal vanished. New stones are a must, otherwise you get uneven bores top to bottom)
4. Install rings correctly (right sides up and gaps clocked according to ring instructions or your favorite way of clocking rings.
5. Don't dip pistons/rings in motor oil when reassembling and don't get any assembly lube on them. Squirt heck out of them with WD-40. Light oil cuts friction but lets the rings seal. You'll get motor oil soon enough from the crankcase.

Can't tell you how many kids I've talked to who re-ringed their motor by taking it apart, seeing smooth bores and just putting in new rings, pistons or both and throwing it back together. No measuring, no bore or hone.

Do it right, guys. It pays off in the end and makes Honda enthusiasts look like they know what they are doing (and care). I know money can be a problem at times, but let it smoke for a couple more months while you scrape together the bucks to do it right.

Unless you've got a date with Jessica Alba, there isn't anything better you can do with that money.