You may be putting too much of your weight on the handlebars. I currently have a rigid MTB (with rigid fork) for commuting and short errands, with 700x42 tires at about 60 psi and I can ride it for longer than 2 hours without hand or wrist pain. I ride on rough asphalt with a few cracks and potholes here and there. Conversely, If I ride my MTB for 2 hours on the flat with no special adjustment, I get wrist soreness.
The differences against my regular MTB for actual mountain riding (Uphill and downhill) is that on my commuter the saddle is a little bit closer to the handlebar and the handlebar is a little bit higher up.
My city is not flat but I definitely ride steeper and longer inclines on the mountains.
Do this test: in your regular sitting position, with both pedals at the same height, try to lift your hands off the bars. You should be able to hover your hands over the grips without having to slide either forward nor backwards on the saddle. My commuting setup is like this, whereas my MTB setup requires me to slide a bit towards the rear in order to do the same.
My reasoning to set up like this is that for commuting I use a lower pace, I'm not trying to beat the clock, I just need to get there, so I use a more upright riding position, a setup more oriented towards comfort. By contrast, my MTB is used more for climbing but I also need more dynamic riding on descents where I need to frequently change my position on the bike. I feel the slightly longer distance between saddle and handlebar allows me to apply higher torque to the pedals for climbing.