Hard to answer. Not all SD adapters support SDHC cards.
If your adapter supports miniSD cards with a capacity larger than 4 GB (miniSDHC cards) then it should also support SDHC cards.
Any device that isn’t specifically designed to support (mini)SDHC can only be used with (mini)SD cards.
I suggest you try to find any specification that came with the adapter or is available on the manufacturer’s site. Failing that, you will have to try and see.