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Two-dimensional gapped semi-Dirac (GSD) materials are systems with a finiteband gap that their charge carriers behave relativistically in one directionand Schr\"odinger-like in the other. In the present work, we show that besidesthe two well-known energy bands features (curvature and chirality), theband-gap parameter also play a crucial role in the index- and magneticfield-dependence of the Landau levels (LLs) in a GSD system. We take themonolayer phosphorene as a GSD representative example to explicitly providephysical insights into the role of this parameter in determining the index- andmagnetic field-dependence of LLs. We derive an effective one-dimensionalSchr\"odinger equation for charge carriers in the presence of a perpendicularmagnetic field and argue that the form of its effective potential is clearlysensitive to a dimensionless band-gap that is tunable by structural parameters.The theoretical magnitude of this effective gap and its interplay with ovalshape $k$-space cyclotron orbits resolve the seeming contradiction indetermining the type of the quantum Hall effect in the pristine monolayerphosphorene. Our results strongly confirm that the dependence of LLs on themagnetic field in this GSD material is as conventional two-dimensionalsemiconductor electron gases up to a very high field regime. Using thestrain-induced gap modification scheme, we show the field dependence of the LLscontinuously evolves into $B^{2/3}$ behavior, which holds for a gaplesssemi-Dirac system. The highlighted role of the band-gap parameter may affectthe consequences of the band anisotropy in the physical properties of a GSDmaterial, including magnetotransport, optical conductivity, dielectricfunction, and thermoelectric performance.

The overlooked role of band-gap parameter in characterization of Landau levels in a gapped phase semi-Dirac system: the monolayer phosphorene case