Short answer:
Since we're dealing with kinetic friction, the force of friction by definition points in the direction parallel to the relative slip velocity between the block and the plane at all times, so there will be no component of the friction force perpendicular to the direction of motion to model.
Long answer:
If you're still reading this response, then I'll assume you'd still like to know how to properly analyze this situation in full detail. I remember analyzing this exact situation when I took a course on Newtonian mechanics taught by Professor Oliver O'Reilly, who wrote and published Engineering Dynamics: A Primer as a supplemental textbook for physics and engineering students studying Newtonian mechanics. Much of what will appear in the following analysis can be found in the primer, but during the analysis I'd like to bring special attention to what O'Reilly has referred to as "The Four Steps" method, which can be implemented to analyze most situations encountered in Newtonian mechanics.
The four steps are:
1.) Define one or more coordinate systems and express the position, velocity, and acceleration vectors for each particle in the system, and state any kinematic constraints on the motion of each particle
Let's choose a Cartesian coordinate system defined by unit vectors $\{\mathbf E_x, \mathbf E_y, \mathbf E_z\}$ such that $\mathbf E_x$ points to the right, $\mathbf E_y$ points into the screen you're currently looking at, and $\mathbf E_z$, which defines the normal unit vector of the surface on which the block moves, points toward the top of this page. For the block,
$$\mathbf r = x\mathbf E_x + y\mathbf E_y + z\mathbf E_z$$
$$\mathbf v = \dot x\mathbf E_x + \dot y\mathbf E_y + \dot z\mathbf E_z$$
$$\mathbf a = \ddot x\mathbf E_x + \ddot y\mathbf E_y + \ddot z\mathbf E_z$$
are respectively the position, velocity, and acceleration vectors. Since the block is constrained on the horizontal surface, $\dot z = 0$ and $\ddot z = 0$. Also, since the surface itself is stationary, the slip velocity $\mathbf v_{rel}$ of the block relative to the surface is $\mathbf v$, simply the velocity of the block.
2.) Draw a free-body diagram (FBD) for each particle and label the forces acting upon them
Using the first diagram in your original post as our FBD, let's identify all of the forces acting upon the block. You have already labelled two forces: the applied force $\mathbf P$, which, in its most general form, is
$$\mathbf P = P_x\mathbf E_x + P_y\mathbf E_y + P_z\mathbf E_z$$
and the force of kinetic friction $\mathbf F_f$, which by definition is
$$\mathbf F_f = -\mu_k |\mathbf N| \frac{\mathbf v_{rel}}{|\mathbf v_{rel}|} = -\mu_k |\mathbf N| \frac{\dot x\mathbf E_x + \dot y\mathbf E_y}{\sqrt{\dot x^2 + \dot y^2}}$$
where $\mathbf N$ is the normal force of the surface acting on the block. Speaking of which, let's add that normal force into our FBD:
$$\mathbf N = N\mathbf E_z$$
The fourth and final force we need to consider is the weight of the block,
$$\mathbf W = -mg\mathbf E_z$$
3.) Use Newton's 2nd Law to construct the equations of motion (EOMs) for each particle
Newton's 2nd Law states that,
$$\mathbf F_{net} = m\mathbf a$$
where $\mathbf F_{net}$ is the sum of all forces acting on a particle with mass m, and $\mathbf a$ is the acceleration of the particle. Let's add up the forces from our FBD and break Newton's 2nd Law into 3 equations -- one for each basis vector of $\{\mathbf E_x, \mathbf E_y, \mathbf E_z\}$:
$$P_x - \mu_k N \frac{\dot x}{\sqrt{\dot x^2 + \dot y^2}} = m \ddot x \tag{$\mathbf E_x$}$$
$$P_y - \mu_k N \frac{\dot y}{\sqrt{\dot x^2 + \dot y^2}} = m \ddot y \tag{$\mathbf E_y$}$$
$$P_z + N - mg = 0 \tag{$\mathbf E_z$}$$
4.) Perform mathematical analysis on the EOMs, and introduce any other fundamental laws of physics that may be relevant to the analysis
We see from the equation for $\mathbf E_z$ that
$$N = mg - P_z$$
Plugging this into the equations for $\mathbf E_x$ and $\mathbf E_y$ and then rearranging, we end up with
$$\ddot x + \mu_k \left(g - \frac{P_z}{m}\right) \frac{\dot x}{\sqrt{\dot x^2 + \dot y^2}} = \frac{P_x}{m}$$
$$\ddot y + \mu_k \left(g - \frac{P_z}{m}\right) \frac{\dot y}{\sqrt{\dot x^2 + \dot y^2}} = \frac{P_y}{m}$$
a set of coupled, second-order nonlinear ODEs of functions $x$ and $y$
, with boundary conditions $\mathbf r_0$, the initial position of the block, and $\mathbf v_0$, the initial velocity of the block.
This is the point where I would go find my friendly neighborhood computer programmer, who would then kindly do all the work of writing out a code that would numerically find approximate solutions to these ODEs for me. However, if you're more experienced with coding than I am, feel free to write up a code for the equations and try specifying various values for the vectors $\mathbf P$, $\mathbf r_0$, and $\mathbf v_0$, and see what you come up with. To verify your code is working properly, make sure to give the code some basic test examples such as
$$\left(\mathbf P, \mathbf r_0, \mathbf v_0\right) = \left(\mathbf 0, \mathbf 0, v_0 \mathbf E_x\right)$$
$$\left(\mathbf P, \mathbf r_0, \mathbf v_0\right) = \left(mg \mathbf E_z, \mathbf 0, v_0\mathbf E_x\right)$$
$$\left(\mathbf P, \mathbf r_0, \mathbf v_0\right) = \left(\mu_k mg \mathbf E_x, \mathbf 0, v_0\mathbf E_x\right)$$
which all result in fairly simple closed-form analytical solutions to the ODEs.
