Training a model based on polynomial regression and then putting that into a linear regression, vs just using the linear regression
There's no one answer about how you should/shouldn't do it.  There are two perspectives out there:

If it works, go for it. One perspective is: if it gives good results, that's all that matters.  So, if you get more accurate regression, great for you.  Try a bunch of approaches and pick whatever works best (taking care to avoid overfitting).

Model the underlying phenomenom. The other perspective is: choose a model that represents how the underlying physical process works.  So, from this perspective, we can't say what you should do, since you haven't told us how the underlying process works; we have no basis for evaluating whether that's a reasonable way to model reality.

There are many possible models one could imagine that map 4 continuous features $x_1,x_2,x_3,x_4$ to a continuous output $y$.  You could use linear regression:

$$y = a_1 x_1 + a_2 x_2 + a_3 x_3 + a_4 x_4 + b$$

Or you could use polynomial regression:

$$y = p(x_1,x_2,x_3,x_4)$$

where $p(\cdot)$ is some multivariate polynomial in 4 variables with degree at most $d$ (for some $d$ you choose).  Or you could use what you're doing:

$$y = a_1 p_1(x_1) + a_2 p_2(x_2) + a_3 p_3(x_3) + a_4 p_4(x_4) + b$$

where $p_1(\cdot),\dots,p_4(\cdot)$ are univariate polynomials with degree at most $d$.

Or you could fit some other nonlinear function (see nonlinear regression), or  fit a piecewise linear function, or use cubic splines, or use kernel smoothing or LOESS regression (see local regression).  There are so many options.  We can't tell you which to use from the information you've provided.

Advice. Usually, a good rule of thumb is that if you understand the underlying process well, use a model that matches the underlying reality.  If you don't know the underlying process, try multiple approaches and see which works best (using cross-validation to avoid overfitting).