The $B$-field is a massless excitation present in the NS-NS spectrum of the superstring. It is a spacetime 2-form. What is generally called $H$ is the corresponding field-strength : $H=dB$, it is a (spacetime) 3-form. In the simplest situations, the condition $H=0$, i.e. $B$ closed, is a natural constraint on a string background. Nevertheless, in more complicated situations, we can have $H \neq 0$, i.e. a non-trivial "H-flux", defining a "H-twist" of the more common cases with $H=0$.

Remark: I am not sure to fully understand the meaning of the question.