Could the sick child in this scene be Fredo and the illness the reason for his slowness?

The sick child was Fredo

but this is only revealed in the subtitles.

He was the oldest, so that is pretty clear.

No he wasn't. Santino was the oldest, then Fredo and Michael and finally Connie.

Just after Vito kills Fanucci we see him and his growing family (pre-Connie) on the stoop.

There's no indication that this childhood illness affected Fredo's mental or emotional capacities but I suppose it's possible.

The novel indicates that although Fredo was fairly tough, certainly enough to be an active Family member, he never quite matched the intelligence and hard-nosed nature of his brothers.

Additionally, witnessing the assassination attempt on Don Corleone seems to have broken something inside him.

Sonny said grimly, “Freddie is no use to us. The doctor says he’s in shock so bad he has to have complete rest. I don’t understand that. Freddie was always a pretty tough guy. I guess seeing the old man gunned down was hard on him, he always thought the Don was God. He wasn’t like you and me, Mike.”

and later...

Michael shrugged. He had seen soldiers in the same condition on the battlefield. But he had never expected it to happen to Freddie. He remembered the middle brother as being physically the toughest one in the family when all of them were kids. But he had also been the most obedient son to his father. And yet everyone knew that the Don had given up on this middle son ever being important to the business. He wasn’t quite smart enough, and failing that, not quite ruthless enough. He was too retiring a person, did not have enough force.

^{The Godfather - Mario Puzo}