Forward Linear Predictor I just need some advice on a project. I have an unwanted sinusoid that has time-varying frequency (its not sinusoidally varying) which is relatively constant over a small section. I've been advised to use a Linear Predictor in order to deal with it and remove it. 

I just need some advice on how to go about implementing it. I understand the theory, my main issue is with this equation for a forward-linear predictor 

$$
\hat{x}(n) = \sum_{i=1}^p a_i\,x(n-i)
$$

Now, what's the point in estimating the next value of the signal if it's also going to contain the same unwanted sinusoid component? I understand how to use this (finding coefficients using Yule-Walker/Levinson-Durbin)

EDIT: More info, Sinusoid is added on top. I can almost hear the telephone conversation clearly when the sinusoid dips down. I understand that for a small enough section, I could just use a notch filter but of course it's not feasible to do this for each small section manually. I need a way of predicitng it's frequency hence the linear predictor The sinusoid has a significantly higher amplitude than the conversation and it's frequency varies from 200Hz to 2kHz
So, the algorithm I'd propose is the following:


Filter your input signal with a 200 Hz – 2 kHz bandpass filter
Since in that band your sine is the dominant oscillation, a PLL would lock onto that. So, use a second-order PLL to track that wave; there's many PLL implementations that'll output the reference as tone with the correct phase. 
Typically, PLLs lose amplitude information, so you can't simply remove the sinusoid by subtracting it from the audio signal. Instead, you'd use the reference tone to multiply the signal with, which mixes down the original tone to 0 Hz. A low-pass filter (in this case, even a single-pole IIR) might be trivially used to extract the instantaneous amplitude. 
Now that you have the right tone with the right frequency and phase, you can subtract it from the input
To predict how the tone is going to be developing its frequency, you could use a linear predictor, but honestly, a second-order PLL is a linear predictor for phase (a linearly changing phase is a sinusoid; the coefficient of the linear term is the frequency). If you need to work with a sinusoid which might be undergoing a linear change in frequency: third order PLL.


Even more elegant: Use the output of 2. Mix your signal down with this. Apply 0 Hz Notch filter (again, single-pole IIR might do, or other kind of narrow high-pass filter). Shift up in frequency again.