This page details some information about "Help Wanted" quests which may be too bulky for the main page
Item Delivery Quest
Seasonal Crops
There's a 15% chance for the item requested to be a seasonal crop (crops cannot be requested in Winter). Otherwise, a seasonal item is requested.[1]
During first seven days of the season, only particular crops may be requested. Starting from the eighth day, any crop may be requested.
New crops begin to be requested at the start of Year 2, as well as after gaining access to the Desert.[2]
These items can be requested during any day of the season without any conditions. For example, fish exclusive to rainy weather can be requested during any weather, and even in Winter.
Willy or Demetrius may request one of these fish to be caught, depending on the season. The amount of fish requested increases the higher player's fishing level is, and decreases the higher fish's selling price is.[4]
Gold Ore cannot be requested until the player reaches level 40 in the Mines. It is replaced with Copper Ore if such quest is generated, keeping lower amount requested and higher reward.[6]
Notes
Item-Delivery-Quest:
Supposedly, some cooked dishes can be requested too. Specifically recipes known by player, all ingredients of which can be acquired during the current season, i.e. are either in the above Crop list or the Seasonal Items list. I have not seen or heard about the cooked dishes being requested, so there are two options:
1) I've misread the code, or it is modified elsewhere, and something prevents them from being added to the list entirely. Likely.
2) By the time players have access to the recipes, they are sufficiently lategame and don't check the HW board anymore. Plus the odds of the recipe being added and picked are fairly slim too. Still possible, but unlikely.