If you search junk food when sick there's all sorts of info telling you not to eat it, e.g.
5 Foods NOT to Eat While Sick or Battling a Cold
#5 Fast Food and Processed Foods
Surprise, surprise – it should be most obvious to steer clear of fast food and processed foods while sick. Fast food and most processed foods are void of nutritional value; that is, they bring hardly anything to the table in terms of boosting the immune system or amplifying your health. Not only are much of these foods made from genetically modified foods, but they also contain a slew of immune-suppressing ingredients such as high fructose corn syrup, aspartame, MSG, artificial colors, FD&C Red #40, dimethylpolysiloxane, and much more. And despite some irresponsible hospitals serving up McDonald’s to their sick patients, that doesn’t mean these foods are any more valuable.
This surprises me, as I thought (1) the easy available energy in sugary junk food would assist in battling the infection, (2) I don't believe GMO foods are intrinsically bad for you, and (3) any alleged immune-suppressing ingredients would be long-term issues, not short term.
Is there evidence that eating sugary foods when sick is a particularly bad time to eat them?