Continuity mistake: In the first shot of the two girls at the very start, Katniss is stroking Prim's hair. She occasionally places her hand on Prim's head. However, when she does this, her hand is in a different spot in between camera cuts. (00:01:50)
Continuity mistake: Katniss puts her quiver on her shoulder for the very first time. Camera cuts and now it's across her back. (00:03:45)
Continuity mistake: When Katniss shoots the rock to draw the deer out of hiding her right hand is uncovered. Then as she fires at the deer a moment later a leather glove has appeared on her hand. (00:05:00)
Audio problem: When Katniss says "that's the first deer I've seen in a year" towards the end of the line you can see her mouth doesn't move. (00:05:50)
Other mistake: When the children of District 12 are walking to the reaping, just after Prim receives the mockingjay pin, a boy can be seen wearing black and white converse shoes. (00:10:15)
Continuity mistake: When Katniss and Prim are entering the square, Prim panics and Katniss tries to calm her. During that dialog, Katniss' hands suddenly appear holding Prim's face. (00:11:00)
Continuity mistake: In the scene where Effie Trinket pulls a name out of the glass bowl, she opens it and the black tab is on her right. She clears her throat, the camera angle changes, and the black tab is suddenly on her left. (00:14:25)
Continuity mistake: During the reaping, as Katniss fights with the peace keepers to get to Prim, there is an obvious break in the shot. In one shot, she is already pushing the peace keeper on her left off, but in the next shot, she is just beginning to push him. (00:15:00)
Continuity mistake: As Katniss volunteers, the Peacekeeper on her right is stood next to her. We see Katniss from a raised angle and this Peacekeeper is now further away. (00:15:20)
Continuity mistake: When Katniss is volunteering to replace Prim, she struggles with 2 guards and starts to fall down. As such her head ends up lower than both guards' shoulders. Camera cuts and she is stood up again. This is an instant cut and a very bad edit. (00:15:35)
Continuity mistake: Just as Katniss is placed on her spot by Effie after walking up on stage, Effie grabs the microphone just as the camera changes. When the camera changes, she is no longer holding the microphone. (00:15:40)
Other mistake: In the shot where Katniss walks up the steps after volunteering to replace Primrose, Effie's fingers are waving her forward, but on the giant screen visible in the background, her hand is still. (00:16:05)
Continuity mistake: At the reaping, As Effie says "our tributes from District 12" the camera pans from Peeta to Katniss. Katniss has her hair to the side of her left eye. Camera changes and now her hair is in front of her eye. (00:16:50)
Continuity mistake: When Prim and Katniss are talking just after the reaping, Katniss grabs Prim's chin. Camera cuts to behind and her hand has gone. (00:18:50)
Continuity mistake: When Haymitch is asking whats the hurry on the train, he has both hands up in the air (one with his glass in). Camera cuts to an angle behind Haymitch and his free hand, which should be visible in this shot, is not. (00:22:50)
Continuity mistake: When Peeta asks what the plan is on the train, Haymitch is holding his glass by the top of the glass. Camera cuts and now he is holding it at the bottom of the glass. (00:22:55)
Continuity mistake: When Katniss meets Cinna, he has 4 earrings in his left ear and 2 in his right, so the last shot (when he says that Katniss "shouldn't be dressed up in some stupid costume") is clearly inverted. (00:29:25)
Continuity mistake: When the tributes are entering the arena on horse-drawn chariots, the spacing between the vehicles keeps changing from being far apart (in the long shot) to bunched up in closer angles. This is not just because the horses either sped up or slowed down because the distance is always perfectly even between each chariot. (00:30:50)
Continuity mistake: When President Snow says Welcome for the first time he has both hands outstretched. Camera changes and one hand is now lowered. (00:32:40)
Continuity mistake: As the tributes' chariots roll into the stadium, the Capitol citizens are throwing flowers to the tributes as they pass. When all the tributes are gathered at the end of the roadway, an overhead shot is shown and the runway is clean and flower-free. (00:33:15)
Answer: I watched this scene multiple times on YouTube. There is definitely something moving that probably is not supposed to be seen. I'd agree with your assessment that it looks like a hand. Most likely a film crew member is crouched on the floor performing some technical function during the filming. Just what it is impossible to tell. You should submit it as a "movie mistake."
raywest ★