No a profit maximizing firm will not produce at a level of output in short run when market price is less than minimum average cost because equality between market price and minimum average cost indicate the shutdown point. So a firm will never operate at a price less than minimum average cost.

In the given graph –

At Price S, and Quantity M

TR = OS X OM, which is represented by the OMRS

TC = SAVC X OM = OP X OM, which is represented by the OMQP

Profit = TR – TC = OMRS - OMQP

As we can see

TC > TR, so there is loss, which is represented by PQRS

Thus the firm shall stop production whenever Price/AR < SAVC

In short run, at profit maximising level, AC ≥ SAVC