Iterative Blind Sinus Signal Suppression
I assume you do everything in discrete time. If I understood correctly then the observed signal is

$$x(n)=s(n)-A\sin(\omega_0n+\phi)$$

where $s(n)$ is a sinusoidal signal with frequency $\omega_0$ and unknown amplitude and phase. The parameters $A$ and $\phi$ are to be optimized such that the power of $x(n)$ is minimized. In a real-time algorithm you can define the cost function as the sum of $x^2(n)$ over a fixed length $N$:

$$F(A,\phi,n)=\sum_{k=n-N+1}^{n}x^2(k)$$

where $n$ is the current sample. For a steepest descent algorithm you need the gradient:

$$\frac{\partial F}{\partial A}=-2\sum_{k=n-N+1}^{n}x(k)\sin(\omega_0k+\phi)\\
\frac{\partial F}{\partial \phi}=-2A\sum_{k=n-N+1}^{n}x(k)\cos(\omega_0k+\phi)$$

The parameters are now updated in the following way:

$$A_n=A_{n-1}-\mu\frac{\partial F_{n-1}}{\partial A}\\
\phi_n=\phi_{n-1}-\mu\frac{\partial F_{n-1}}{\partial \phi}$$

where $\mu$ is the step size. Note that in every iteration you need to recompute the gradients with the current values of $A$ and $\phi$.

The figure below is a simulation of this gradient algorithm with the unknown signal $s(n)=2\sin(\pi n/3+\pi/2)$, $N=50$, $\mu=0.01$, and with zero initialization of both amplitude and phase (amplitude $A$ in blue, phase $\phi$ in green).