A vortex beam is a kind of beam with a spiral phase surface, and in the beam center there is a phase singularity, thus the intensity is zero. Vortex beams can theoretically carry Orbital Angular Momentum (OAM) with any mode, and the vortex beams of different OAM modes are orthogonal to each other. Because of these special properties, vortex beams have received extensive attention from researchers and have been widely applied in the fields of wireless communication, optical tweezers and microscopes. However, the most researches on vortex beams concentrated on the canonical ones, whose phase gradient is constant along a circular path around the vortex center. The canonical vortices can be simply expressed as $A{\mathrm{e}}^{\mathrm{i}n?}$ ($A$ is the amplitude of field,$?$ is azimuthal angle and $n$ reperents the topological charge of the vortex), which in many circumstances are usually called ‘optical vortices’ for short. The ‘noncanonical’ means that the phase gradient of the vortex is not constant, while the topological charge is also the same as it in its ‘canonical’ counterpart. Actually, the noncanonical vortices were studied by several scientists, but only the forms of the noncanonical vortices satisfying the paraxial wave equation were taken into consideration. There, the ‘noncanonical’ vortices were also called ‘anisotropic’ vortices, since not only the phase but also the amplitude of such vortices was spatially anisotropic. The researches on the noncanonical vortex wave are mainly concentrated in the scalar field, such as the study of the trajectory of noncanonical vortices in free space, the phase singularity and energy flux of noncanonical vortex dipole Airy beams. Recently, PANG Xiaoyan et al proposed a new type of noncanonical vortex, named X-type (noncanonical) vortex. This vortex inherits and develops the conventional noncanonical vortex, i.e., it no longer has a constant phase gradient around the beam center, while the intensity keeps invariant azimuthally. In this article, the X-type (noncanonical) vortex with left-handed circular /right-handed circular polarization is studied in a tightly focusing system. In this tightly focused (vectorial) field, the Spin-Orbit momentum Interactions (SOIs) occur, and as we know that the SOIs play crucial roles in many circumstances and studying the optical behaviors involving SOIs continuous to attract a lot of attention. As it is shown in this article under the SOIs the X-type vortex shows interesting propagation properties. By applying Richards-Wolf diffraction theory, this three-dimensional (3D) focused field is expressed and analyzed. Here we mainly focus on the intensity distribution of this 3D field, and the behaviors of the total intensity on the focal plane and on the transverse planes along the beam propagation are discussed. The topological charge of the X-type vortex in this article is chosen as $\mathrm{1}$, the fundamental charge, and $\mathrm{2}$ the higher order charge. For the much higher orders, such as $\mathrm{3}$ or $\mathrm{4}$, the analyses of these fields can follow the same method that given in this article. The comparison between the behaviors of the field intensity for the incident field with left-handed circular polarization and the right-handed circular polarization are made, which also is discussed with the result in previous study for the linear polarization. It is found that the total intensity of the field shows the transverse focal shift in the focal plane and the rotation behavior along the propagation, and these properties of the intensity are distinguishing from those in the field without SOIs. The main conclusions are: First, the anisotropy parameter $c$ of the X-type vortex is the main reason for generation of the transverse focal shift and the intensity rotation of the X-type vortex, and the semiaperture angle $\alpha$, polarization state and the topological charge all have effects on manipulation of the intensity distribution. Second, both the rotation direction and the total rotation angle of the intensity pattern of the focused field are independent of the polarization state of the incident field. Third, the difference between the intensity distributions of the strong field for the incident wave with the left-handed circular polarization and the right-handed circular polarization is a result of the SOIs, and the same behavior for these two cases is that they have the same switch value $c=\mathrm{1}$on the focal plane, which is different from the switch value of the linear polarization. In the case of linear polarization (without the SOIs) the switch value of the anisotropy parameter is always smaller than or equal to $\mathrm{1}$, i.e., it changes with the system parameters. Fourth, for the second-order of the X-type vortex, although the transverse focal shift and the intensity rotation of the beams are different for the left-handed and right-handed circular polarizations, the distribution of the main vortices and the rotation behavior in two cases are the same. Our work may give a new way to explore the noncanonical vortices and will supply an additional method for tailoring 3D optical fields, which may have potential applications in optical tweezers and other fields.