Not only does he have a cold, but he also has a fever.
What is the function of 添？ I feel rather awkward saying it. I searched the dictionary and doesn’t look like it has added any meaning.
You might be interested in these notes from ABC Cantonese on the word 添:
bei2 do1 ngo5 jat1 baat3 man1 tim1
Give me one hundred dollars more
sik6 do1 di1 tim1
Eat some more
Notes: occurs at end of sentence to indicate action is expected to be repeated; usually pronunciation with high falling tone; also written as 㖭 tim1, 添 tim1
2 as well, also, more, again
keoi5 zong6 dou2 jan4 m4 dou6 hip3, zung6 naau6 jan4 tim1
After bumping into somebody, he didn't apologize, and even scolded that person as well
gam2 zou6 faat3 zung6 bai6 tim1
Doing it this way is also even worse
Notes: follows 仲 zung6 to emphasize something additional is involved
仲 = also
添 = add / additionally
[(仲) ~ (添)] is a set structure for [(in addition, also) ~ ]
仲發燒添 means "in addition, also have fever"
添 can be optional. It's used to exaggerate his expression. You can say