It's been a while since I did my exterior hood vent selection.
One aspect I notice looking at your photo is the pitch of the exterior vent opening doesn't appear to be parallel to the wall but rather appears facing out at you. Wonder if wind might drive rain straight on. Most exterior wall hood blower units have both a bird/insect cage and as mentioned above, a damper somewhere along the run to keep prevailing winds during rains from getting into the duct system and beyond. The link provided above shows the dampers well. Do you know where the damper is? For example, mine is a butterfly type shown above and is above the removable baffles of my vent hood
At installation of the external blower vent, after the blower ring is placed in the wall opening, the blower is attached by six screws to the wall. All required screws on the back panel must be placed, so that any moisture that may get inside the housing won't leak into the house.
Perhaps you may wish to check these two aspects for potential help to your problem. Good luck.