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I’ve noticed today riding slow with my kid alongside- my orange has what I’d call ‘front wheel flop’ at low speed- suddenly flops one side to other even holding bars- quite hard to stop or correct
It's a combination of having a high trail and a front load (even unladen) mean that once it turn very slightly, the weight of the rack increase the turn/flop more, due to the fact it's an MTB frameset with a suspension corrected fork to keep the trail high enough for stability off road.
Solution is basically a shorter fork to steepen the head angle and reduce trail, but seeing how far back your saddle is, you'd need a little more layback to put you in the same position.
Having said that, with the amount of seatpost showing, maybe replace the frameset with an Intec one? if that's 26" wheels, then the Intec M1 Disc.
This probably doesn’t tell you much but this is the floppy bike as it is now!