Rear wheel drop-out adjustment (1-speed)
I would not ride that bike, for brake reasons.

1) The rear rim brakes would be out of position.  Hopefully you've re-adjusted the brake pads.  However rim brakes would be safer than other brakes given that dropout.

2) If you have a coaster brake (ie a backpedal brake)  then a hard stop could tear the axle out the front of the dropout, using the ground as a lever-fulcrum.  

3) If you have a drum brake (which is possible given I can see a cable outer and the axle is hollow, perhaps its just not connected in the photos) then the same as for backpedal brakes.

Your safe answer is to simply buy and fit a longer chain.   Yes its annoying.