My cat is listless, always lying down, cries when touched, and often has (sometimes yellow) eye discharge.

From the picture the cat looks like a Scottish Fold; if it is a Fold that always lies down and cries when its limbs are touched, suspect an attack of inherited cartilage disease. This is only a preliminary judgment from your description; a vet must confirm the cause. Fold cartilage disease is abnormal cartilage development with a very high onset rate, especially in poorly bred folds.

It shows as swelling of the tail, knee, and ankle joints, with whole-body pain on attack causing movement difficulty and crying when touched, and inflammation raises tear production. For an affected cat, take it to a vet for examination and use NSAIDs for pain relief plus long-term chondroitin supplements..