Build barracks in your major core-province cities. At level 3 they generate 50% more manpower.

Also check on morale, the higher the morale, the more provinces produce.

Yes.

I'm not sure if building other constructions increases the base population of a province, and therefore the manpower production, anyone know?

no. barracks just effect man power and the base population can not be changed. Not by Moral and not by any building in the province.

Thanks for that clarification.

The manpower in my capital city has actually dropped, from 1715 per day to 1664 on day 3 - and AFAIK morale has only been increasing in that time. Anyone know what might have caused the drop?

Update: Looks like my manpower levels have dropped everywhere. Why would this be happening when the province morale is increasing?

I guess it could be that manpower levels are determined by your country's overall morale rather than province morale.That would be one possible explanation, as I conquered three new provinces recently.

your units are useing manpower for upkeep. that reduces effective production

Yes, my apologies, I realized that after posting. I thought manpower upkeep would come from your stockpile, but it doesn't