The propagation of interface waves at the interface between a fluid-saturated porous medium and a fluid has been extensively investigated in the last three decades due to its various and wide applications in several fields including earthquake engineering and materials testing. Although the sea floor is usually covered with porous marine sediment, the previous interface wave theories are rarely used for submarine acoustic problems for the following reasons. 1) In addition to hard porous media, unconsolidated soft porous media exist widely in the seabed, which are seldom considered in previous studies. 2) The depth of seawater is limited, and in many cases it cannot be regarded as a half-space. 3) The fluid-saturated porous medium model cannot describe the effect of a small number of bubbles caused by decomposition of organic matter in the sediment. Hence, the present paper focuses on the low-frequency pseudo-Scholte waves at the interface between an overlying fluid layer of finite thickness and a quasi-saturated porous half-space. The overlying fluid is assumed to be ideal compressible water and the quasi-saturated porous media are assumed to be sandstone and unconsolidated sediment and modeled by Biot theory. A fluid equivalent model is used to analyze the effects of the bubbles in the pores. Based on the boundary conditions, the closed-form dispersion equations of far-field interface waves are derived by using classical potential function method. The velocity and attenuation of pseudo-Scholte wave are determined by Newton iteration in a reasonable rooting interval. The analytical expressions of the displacement field and fluid pressure distribution caused by pseudo-Scholte waves are also derived. Then, based on the derived theoretical formulation, the numerical examples of calculations are presented. Our calculation results show that the stiffness of porous medium significantly affects the mode, phase velocity, displacement and fluid pressure distribution of interface waves; the phase velocity of the pseudo-Scholte wave in the finite-thickness fluid/fluid-saturated porous half-space is related to the ratio of the wavelength to the thickness of the fluid layer; the phase velocity of the shear wave is insensitive to a small number of bubbles dissolved in the pores, but the existence of bubbles has a significant influence on the phase velocity of the compressional wave and the pseudo-Scholte wave. Furthermore, the existence of bubbles can significantly affect the distribution of the pore pressure.