We report on a type of 2 μm h-shaped pulse generation in a thulium-holmium-doped fiber laser and experimentally investigate its cavity birefringence and pump power dependences. An asymmetric nonlinear optical loop mirror is employed as an artificial saturable absorber, which incorporates ∼52.7 m dispersion-shifted fiber and ∼3.8 m ultra-high numerical aperture fiber to enhance the nonlinearity. The h-shaped pulse shows both a polarization state (PS) and pump power related evolutions, even when randomly weak birefringence fibers are employed. By further incorporating different lengths of high birefringence polarization-maintaining fiber (PMF), i.e., introducing different amounts of linear cavity birefringence, much larger pulse tuning ranges can be realized. In particular, when the PMF is lengthened to ∼2.3 m through manipulating the PS, the achieved longest pulse duration of ∼318.14 ns can almost cover the whole repetition period of ∼323.96 ns, corresponding to a pulse duty circle of ∼98.2%, the largest ever reported from a fiber laser, to the best of our knowledge. We demonstrate the related characteristics in detail.