motor mounts are built in several variations.
1 .rubber / steel solid mount.
2. combination of coil spring and rubber dampner,,these are not used any more.
3. combination rubber and liquid filled isolator cell, known as liquid filled engine mounts.
Engine mounts are attached to the vehicle chassis and directly to the engine,,or mounts are used in conjunction with other types of subframe mounting apparatus.
Generally while the subframe or the mounts attachment points may be tightened, there is usuall no provision made for the mount itself to be tightened.
Lets start from the beginning :
You said that a state inspector reported that the motor mounts were loose.
The state inspector, while a very knowledgable individual uses general inspection criteria to make a determination.
The ability to make such a decision in no way makes the inspector a honda certified mechanic.
So,,here we are,,other than the inspector,,nobody has done a visual inspection of your car,,not the non-honda mechanic,,,or the honda dealer,,,,
Take your car to both of them,,tell us what the inspection revealed,,including repair cost and parts necessary,,,then we can go from there.