Sometimes "small" missions get their mood up, but I've not found how exactly.
Yeah, this is kinda wonky/glitchy too. Basically, if you send someone out on a mission and they don't die (success or failure) then you will lose mood. However, if the NPC you sent out is below average skill in his/her profession, the mood loss is cut significantly. Similarly, if the NPC you sent out is below average in missions (guaged only by successes: take note of this, its important) completed, the mood loss is again cut significantly. If the NPC is below average by both metrics, you will gain a small amount of mood instead.
Now, to game this, you basically need to find an NPC who is an adventurer by profession before
building any adventuring guilds (you want his/her skill level as low as possible), then boost his/her mood up to the point of 100% survival on the safest arcane ingredient quest (rather trivial). Next, get a different NPC up to a handful of arcane ingredient dispatch successes, and then give a single skill book to any other adventurer. Now, your first adventurer is below average in both stats, and ideally has a near-zero success rate and a 100% survival rate. Now just keep sending him/her out on arcane ingredient quests and with return, his/her mood will increase and because you're failing each dispatch, the NPC still remains below average in skill and disptach successes. You can hypothetically do this a few times to prop up a few NPCs by just layering more a skill book on to each "primed" adventurer.
Once done, finally build your adventurer's guilds to make them actually functional.
At present, you don't actually need to get the skill level down that low, if the skill level gives you a success rate of about 50% or so you'll still endlessly accumulate mood AND gain arcane ingredients provided you stay on the NPC menu
. However, this is likely a bug that'll soon be fixed as it relies on the dispatch stats failing to update after a given dispatch (rather, only updating when you return to the main dispatch menu).