Is there an efficient algorithm to extract low frequency signal from audio with high frequency carrier?
If you know your carrier is always exactly 16kHz (within a few percent or so), and you have sampled the audio at 48kHz, you can try a simple 3-tap FIR filter approach:

$$Smoothed(k) = \frac{1}{3}( Sig(k) + Sig(k-1) + Sig(k-2) )$$

This assumes your original signal is called $Sig()$, and it computes a new signal, $Smoothed()$.

The method exploits the fact that three sample periods equates to one entire cycle of a  16kHz sine wave, so an 16kHz component in the input signal will be cancelled (since summing all the samples of a full sine wave cycle will result in zero).

The resulting $Smoothed()$ signal should look like you original plots, with the furry 16kHz components gone.

If your sample rate is not 48kHz (let's call it $F_s$), or if your interfering sine wave is at some other frequency (let's call it $F_i$), you can craft the 3-tap filter as follows:

$$Smoothed(k) = \frac{1}{2-2\cos(\omega)}( Sig(k) - 2\cos(\omega) Sig(k-1) + Sig(k-2) )$$

where $\omega = 2\pi\frac{F_i}{F_s}$ (note that, if $F_i=16000$ and $F_s=48000$, we get $\omega=\frac{2\pi}{3}$ and hence $\cos(\omega)=-\frac{1}{2}$. If you substitute this into the second equation, you will simply get the first equation again.

This method is simply creating a $2^{nd}$ order FIR filter with zeros at $\pm F_i$, and with the gain at DC normalised to unity.