Answer: Crickets do eat plant parts. I suspect they are the culprit you are looking for. I am not aware of an organic control for crickets. I would suggest you pull the mulch back away from the plants as it is obviously harboring the pests. This may be enough to get your begonias through without the need for spraying a pesticide.
Mole crickets are primarily feeders on the roots of turfgrass. They are very different in appearance from other crickets. They sort of remind me of a stocky, long cricket with lobster-like front claws. I don't think they are your culprit.
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