If you have the item, it will usually say 1 under the quantity on the right instead of 0. You can check if you have it in retainers and the saddlebag by clicking "search for item" on the item's menu.

I don't know which 3 raids you are spamming but the most poetics come from Castrum Meridianum, Aetherochemical Research Facility, roulettes and S ranks. You can help someone through all 12 alexander raids for the first time for 1,200 poetics or queue for any old ARR trials such as Howling Eye (Hard) and if it's someone's first time you get a bonus.