For Cinccino's "Make Do" Ability, can I use it when I'm discarding a card as part of some action or effect?
No. Discarding a card from your hand is part of the cost of using the Ability. You can't use a discard done from some other effect or game action.
Source: Sword & Shield FAQ (2020-01-24), TPCi Rules Team (2020-02-06)

Relevant cards

Note: All applicable cards may not be listed below.
SWSH1_147
SWSH1_147
SWSHP_SWSH009
SWSHP_SWSH009