What is the best way for an asthmatic child to inhale when using the nebulizer machine?

Breathe normally. The nebulizer spreads the medication dose over about 5 to 10 minutes. The child should breathe normally through the mask attached to the machine. Tip- if the child is sitting upright so that the medication cup is upright, the whole system works better.
AN addition. An additional point is that "blow by" that having the mist from the nebulizer blow by the face of the patient does not effectively deliver the medication. Also, a screaming child will have very little medicine delivered via the neb to their lungs -- in fact most will end up in their stomach. These are two of the most common myths and mistakes made in using a nebulizer.