Let $M$ be a smooth manifold ($M=\mathbb{T}^2$ is enough for my purposes) and $f,g:\mathbb{S}^{1}\to M$ two smooth loops, such that $f(1)=g(1)$ and $$\left.\frac{\text{d}}{\text{d}t}f\left(e^{2\pi it}\right)\right|_{t=0}=\left.\frac{\text{d}}{\text{d}t}g\left(e^{2\pi it}\right)\right|_{t=0}. $$ If $f$ and $g$ are homotopic relative to $\{f(1) \}$, then it is possible to prove that there is a smooth homotopy $$H:\mathbb{S}^1 \times[0,1] \to M $$ such that $H(s,0)= f(s)$, $H(s,1)=g(s)$ and $H(1,t) = f(1)=g(1)$.
My Doubt: Is it possible guarantee that there is a smooth homotopy $\tilde{H}: \mathbb{S}^1 \times [0,1]\to M$ between $f$ and $g$ (relative to $\{f(1)\}$) such that $$\left.\frac{\text{d}}{\text{d}s}\tilde{H}\left(e^{2\pi is},t\right)\right|_{s=0} = \left.\frac{\text{d}}{\text{d}s}f\left(e^{2\pi is}\right)\right|_{s=0}, \quad\forall \ t\in [0,1] ?$$
I was trying to demonstrate the above possible result using a local chart and "adjusting the slope" of $\frac{d}{ds}H(s,t)$, but I wasn't able to do such thing.
Solution when $M=\mathbb{T}^2$
If $M = \mathbb{T}^2$, then
$$p: \mathbb{R}^2 \to \mathbb{R}^2 /\mathbb{Z}^2 $$ $$x\mapsto [x] $$
is a covering map and a local diffeomorphism.
Lifting the loops $f,g$, we find the functions $\tilde{f}, \tilde{g}: [0,1] \to \mathbb{R}^2$, such that $$p\circ \tilde{f} = f,\ \text{and}\quad p\circ \tilde{g} = g. $$ Once $f$ and $g$ are homotopic, $\tilde f(0) = \tilde g(0)$ and $\tilde f(1) = \tilde g(1),$ moreover $\tilde f ' (0) = \tilde g ' (0)=\tilde f ' (1) = \tilde g ' (1)$ (we are taking the derivative in the extended sense, considering $[0,1]$ as a manifold with boundary).
Then, we can define \begin{align*}H:&[0,1]\times [0,1] \to \mathbb{T}^2\\ &(s,t) \mapsto p \circ ((1-t) \tilde f(s) + t \tilde g (s) ) , \end{align*} using that $H(0,t) = H(1,t)$ and $\frac{\text{d}H}{\text{d}s} (0,t) = \frac{\text{d}H}{\text{d}s} (1,t)$, we can induce the smooth homotopy ( where $\mathbb {S}^1 = [0,1]/\sim$)
\begin{align*}\tilde{H}:&\mathbb{S}^1\times [0,1] \to \mathbb{T}^2\\ &([ s],t) \mapsto p \circ ((1-t) \tilde f(s) + t \tilde g (s) ), \end{align*} which satisfies the requested conditions.
