4 month old wants them, I would give them. My pedi was pro food at an earlier age than most. Rice Ceral is a safe bet as others have already said, it's filling and very easy to digest.
This is what worked for me with my little guy: I had a very hungry boy, I fed him during the day every 2 hours, so there would be 5 feedings in a period where you now do 4, maybe do 4 or 5 ounces each instead of 6, then give him a little rice cereal, once a day to start then twice after he's had time to adjust...see how he does with it. Then this is where I got lucky, he was able to eat so much during the day, that at night as soon as I knew he could sleep 6 hours straight,,, I would feed him right before I went to bed (even if he was already sleeping) i picked him up gave him a bottle and laid him back down. then he slept the next 6 hours, so didn't have to get up at 2 am anymore.
My Pedi also stressed in no uncertain terms to NEVER put rice cereal in a baby bottle <sure tons of people have done it successfully> but he subscribes more to thats where childhood obesity can start to kick in, because we really have no way of knowing exactly how much they are getting, once it's diluted it all changes..and the purpose is to teach them to eat, the rice in the bottle defeats that purpose.
Good Luck!! Hope this helps