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I'm sorry to hear your cat isn't feeling well. It may be difficult to treat at home if he continues to cough or regurgitate this slimy saliva. However, you can try, as long as he isn't too lethargic, pale, unresponsive, or painful. It may just be some stomach upset, but if not clearing up soon, you'll want some diagnostics from your vet to rule out problems like infections, liver or kidney issues, parasites or asthma, to name just a few.
Wait at least 12 hours before feeding, a bit longer if still coughing anything up. Then you can try very very small amounts of a bland food, like baby food (chicken or turkey flavors are good) or boiled chicken. Just about a tsp or bit less.
Try again in an hour this same amt. If doing ok, do this for a few hrs, then about 1/2 a meal amount for his next meal time. Even though he hasn't vomited anything like food, his stomach could certainly be causing this upset, and this is worth a try.
If doing ok, keep on bland diet for a couple of days, the gradually switch back to normal food. If not able to hold down anything down, he'll need a vet visit, as diagnostics will be called for. Also, make sure he's drinking water the entire time so as not to dehydrate, but only small amounts at a time offered, and keep him indoors while recuperating.
Best of luck, hope your cat is better in no time!