Much of your sample was simply flare from the sun directly entering the lens aperature. There is no solution for that... blocking it will also block part of the image. But just before the end at about 1:05 on the timeline, there is a white "foggy" flare in the lower left corner. That is flare caused by off-axis overhead sky light reflecting off the side of the hole through the case and into the lens. It can be diminished by enlarging or tapering the hole in the case, making it wider at the front outside edge (cone-shaped). But this requires removing the entire camera from the case. Trying to bevel the case opening without removing the circuit board and lens could easily scratch the lens. You camera focus is dead-on perfect... I'd be concerned removing the circuit board could change the focus or even damage the ribbon cable if the lens is hot-glued to the case.
Alternatively, you could make a lens hood that would attach to the outside of the case and block that off-axis light from hitting the hole on the case, but attaching it requires glue, and then the case can't be opened for battery replacement with out removing the hood. And finding just the right size and shape of hood to block the unwanted light yet allow the light needed without blocking part of the recorded image or increasing the vignetting is no easy trick.
I've done both of these solutions. The first is easier by far, but it's easier yet to just enjoy what this tiny camera can do and live with the occasional flare. I've not seen a sharper, brighter, better color video from the #16 camera than the one you posted!