Short vowels are often mispronounced because they are a small part of language learning many people don’t bother to learn proper pronunciation until they experience the confused responses of their listeners.
Short e is pronounced [ehh, ehh] as in egg, leg, beg. The lips are lifted by the right and left corner of the mouth. The mouth pulls both corners up slightly as if beginning to smile.
At the same time air is released from the bottom of the throat and pushed out between the upper and lower teeth producing a very soft air stream. This is how [ehh] is produced.
My upcoming post will guide you through an exercise that will make the correct sound a habit.