The Hopf Umlaufsatz states that the angle of the tangent along a simple smooth closed plane curve turns by +360 or -360 degrees. See curve (1) in the following picture. The Hopf Umlaufsatz can be proved by the deformation trick. See this page. For the sketch of proof, see pages 27-28 of Professor Theodore Shifrin's book.
A plane curve is a map $c:[0, L]\to \mathbb{R}^2$. If $c(0)=c(L)$, then the curve is closed. The curve is simple if for every $0<s_1<s_2<L$ we have $c(s_1)\neq c(s_2)$. If there are $s_1$ and $s_2$ satisfying $0<s_1<s_2<L$ and $c(s_1)=c(s_2)$, then $s_1$ and $s_2$ are called the self-intersecting points. See the following picture.
For a curve with self-intersecting points, curve (2) in the above picture, the Hopf Umlaufsatz is not applicable. Which line of the proof in the book would not hold?
Intuitively, the angle of the tangent along curve (2) turns by +720 or -720 degrees. Can we use the deformation trick to prove this statement?
It is easy to answer your question: The chordal map $\mathbf h$ is not defined when $s=s_1$ and $t=s_2$.
If you want to analyze this further, I suggest you look at (and prove!) Exercise 13, which gives a generalization of the Hopf Umlaufsatz to the case of a piecewise smooth curve; you can write your non-simple curve as the union of two curves, each with a corner point.