OMG--the LAST one is FUNNY!!!!! I love that! That shot is going to be a special family memory in the future. It just captures that new baby time frame, where an infant is sort of...I don't know how to put it..."new to the world and its myriad strangeness," I guess. The baby looks so bewildered. I love it! From a technical standpoint, it does show one thing--the need for a good lint brush for every black-shirted shoot you do.
I recall very clearly my first white backdrop + gelled flash experience in 1986...wow...I had much,much worse blowback than you got, but I was using a powerful studio flash...I ruined most of the whole shoot. White fabric or walls reflects a lot of light. Also, I can see b y the shadow outline on the backdrop in shot #1 and #4 that the people were fairly close to the paper, so they have attachment shadows from the main light.
In #1, the dad hip-holding the baby looks awkward. I think I would have given Mom the infant. In #4, I might have given Mom an apple box under her right foot, to help elevate the infant a bit higher, put a foam block with cover under the baby's butt, to elevate him that way.
If you had fired the speedlight onto black, or thunder gray paper, it would have dramatically decreased the reflection of the gelled light on to them. I know space if often at a premium. One thing too--if yuo open blue gel backdrop images in POS, and go to selective color, you can adjust the backdrop color pretty easily.