This communication introduces a modified substrate integrated waveguide (SIW), the gap SIW, and deduces an approximate closed-form expression for calculating its effective width, taking into account the fringing and coupling fields around the gap. As an immediate application, this new structure is used to correct the phase distribution of an H- plane SIW horn antenna. The achieved quasi-uniform distribution results in a greatly enhanced antenna gain without deteriorating the bandwidth. To further improve both radiation and impedance matching, printed tapered-ladder transitions are introduced in front of the horn aperture. Software optimization confirms the possibility of simultaneously achieving an excellent phase correction (and hence increased gain) and a good impedance matching in a probe-excited horn antenna model. Measurements on a homemade prototype validate the design strategy. The final compact Ka-band planar horn antenna achieves an enhanced gain of 10.3 dBi at 34 GHz, with an impedance bandwidth of 20%.