I'm sorry to hear your little kitten is not doing so great. It's reassuring to know he has been assessed by the vet, however it sounds as though he is taking steps backwards rather than forwards right now.
When a kitten is taken away from their mother at a young age, it is not unheard of for them to struggle to cope as they may not have completely finished weaning and are still somewhat reliant on their mother. Having said that, the majority of kitties who leave mum too early go on to do just fine.
The probiotics and wormer should be continued as will help to address any internal parasites and should solidify the stool.
Young kittens have very low body fat and water reserves so the risk is that they can become dehydrated very quickly if not eating properly or if suffering with diarrhoea. As your little one is now showing signs of lethargy, this may be a sign of dehydration. You might also notice his gums feel sticky and dry rather than wet and slippery.
As kittens can go downhill quickly, I would want this little man re-assessed by a vet immediately. He likely needs some form of rehydration (whether it is via an oral rehydration solution, subcutaneous fluids injections or a drip) and we need to get him back to eating 3 to 4 meals a day. The vets should have some syringe feeding options for you so you can make him eat. Only eating once a day is nowhere near sufficient for a 7 wk old kitten and he will start to lose weight soon.
With sneezing and ocular discharge he may also have a mild upper respiratory tract infection. These kitties can feel under the weather and may have a mild fever. If their nose is stuffy they cannot smell their food as well, so may not have a great appetite. This is another reason why rehydrating him and making him eat is essential. If he does have a fever, the vet can give you some medicine to bring this down.
You could try to encourage him to eat with warm chicken/white fish or a warmed up wet kitten food. Warming it up can help to release the tastes and aromas. Offer it on your finger. You can also mix warm water in, to encourage more drinking. However, it sounds as if he is at the stage where he may not take to this and will need some additional vet help.
While kittens can become poorly quickly, they can also recover just as quickly. Fingers crossed with just a day or two of treatment he will begin to thrive once again.
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