It seems that I can not use together
- fleqn option [to move the equations to the left] and
- \leqno [to give a hand-made label to equations].
This seems to work fine:
The equation number is on the left and the equation is left aligned for all display mode equations.
If you don't want all equations left aligned, then ydon't add the package option
Just saw the reference to hand-made labels in your question. If by this you mean custom labels, then you can adapt the solution from Label equation with a symbol as follows: