When I play gust monk can I pay the additional cost of there is no cards in the discard

Yes — you can pay the [1] additional cost, but the Assault 2 effect won't happen.

The additional cost to play Gust Monk is only 1 Energy — it doesn't require banishing anything. So an empty trash (discard pile) doesn't stop you from paying it.

The problem is the trigger: "banish a card from any trash to give Assault 2" uses cost-within-instruction wording ([do X] to [do Y]). For triggered abilities, that cost must be paid when the trigger is placed on the Chain — if no cards exist in any trash, the trigger can't finalize and is removed from the Chain (Rules 740.4.a.2, 404.2). Gust Monk still enters play normally; you just waste the 1 Energy and grant no Assault 2 (compare FAQ #5052).