When we leave an offering for the God and Goddess, it is generally something that is appropriate for the specific situation. When I am working with Kali, I keep a bowl of jasmine on my altar with honey in it because jasmine and honey are sacred to her. When I'm working with Isis and Osiris, I usually burn handmade kyphi incense on my altar because kyphi is an ancient ritual incense the Egyptians used to use. If I am working with Pan or the Greenman, I have a bowl of willow and oak bark on my altar. For the healing goddess, I would have a bowl of St. John's wort and lavender.
If you want to have an offering to specific deities, you want to make sure you do your research and use items sacred to them. The best time to leave an offering is intuitive, whenever it feels right. Personally I put the offering on my altar, leave it for a few days, and change it out.