Forward Linear Predictor
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.