Interpolation for rational curves with secants

In arbitrary characteristic, we determine the maximum number of general points through which a rational curve of degree $d$ in $\mathbb{P}^r$ passes, subject to an additional secancy condition along a linear space. We consider the cases both where the points on the curve are unprescribed and prescribed, which amount to the determination of the normal and restricted tangent bundles of a general rational curve in $\mathsf{Bl}_{\mathbb{P}^s}\mathbb{P}^r$, respectively. In the appendix, we enumerate the interpolating curves in the case of prescribed points on the curve.