One popular method that I came across for interpolation of a set of points is by using cubic Bezier curve segments with $C^1$ and $C^2$ continuity conditions at the junction point (or node) between two segments. The result is a set of Bezier curve segments that are smoothly connected. Is there a way to be able to control how much these segments are close to the piecewise linear function connecting every two consecutive nodes? I know there is something called tension spline that can follow such piecewise linear interpolation by changing a tension value, but can we do the same thing with Bezier interpolation?
You seem to know about splines with tension controls. Some of these (the Alan Cline variant) use exponential functions. Others, like the $\nu$-splines developed by Greg Nielson in around 1973, are piecewise cubic polynomials.
Since any piecewise cubic can be expressed as a string of Bézier curves, the short answer to your question is “yes”.