influence is a per-vertex property. when you delete some geometry (mesh vertices) and replace it with new geometry (mesh vertices), those new come without influence assigned. the game engine does not know how to move them softly when player/ped plays an animation as vertex-to-skeleton assignment is not available on newly-added vertices.
If you've added vertices to the mesh already, you can copy influence within this mesh only. select (select\element) unweighted vertices, pick weights copy tool and click on any of selected vertices -> these will copy influence onto each of selected vertex using the nearest properly-weighted vertex, assuming "this new will act the same way as this good one". The drawback of the situation is that original "task" mesh has no enough "nearest" vertices to copy good influence from. that is why I suggest to copy influence before attaching the mesh, and copy it from "upper" mesh model.
make sure to set Properties->Mesh->Vertices->Format->MaxRigBones = 4 onto your object before you copy influence weight onto.
take a look at this old video. it has this "copy" technique shown on:https://youtu.be/BGj0szxzeWM?t=386
...attaching a mesh (made of cylinder) to a weight-mapped ped model.
(the most important starts at 11:15, but watch the full from 6:30)