The frequency of a signal is hard to change. Some ways that you can change it:
- reflect it off a moving object
- change the distance between source and receiver as a function of time (Doppler shift)
- introduce some non-linear amplification - this will generate harmonics
If you just add "noise" (uncorrelated to the signal), you can create constructive or destructive interference - but on average you will not change the "recorded frequency". Of course if your "noise" consisted of a single tone equal to the sound of interest but of opposite phase, plus a second tone at a different frequency, then the sum of signal plus noise would be a tone "at a different frequency than the original". But I would hardly call that mixed-in signal "noise" - it would be more properly called a carefully constructed jamming signal.
There is the possibility that you meant to ask about the amplitude of the signal. Again, if the signal you mix in is random, uncorrelated noise then it will at times contain the frequency of interest - but with a phase that is random compared to the signal. So sometimes the two would add, and sometimes they would subtract. On average there would be no difference.
A third possibility of the mixing of signals: if you have a pure tone at one frequency (say 440 Hz) and you add a second tone of a slightly different frequency (say 442 Hz), then you will hear a "beat" in the signal at 2 Hz (the difference) and may also think that you are hearing the sum frequency (842 Hz) and the "mean" frequency (441 Hz). That doesn't mean those frequencies are "there" - but the ear is good at making things up. You see this in Fourier transforms - if you have an FFT with frequency bins of 440 and 442 (bin spacing 2 Hz - for example a 1 kHz sampling rate, sampled for 500 samples) then if you sampled a 441 Hz signal you would "see" 440 Hz and 442 Hz signals in equal amplitude in your FFT. You could interpret that as "two tones close together" or "a single 441 Hz tone". There is no way, with a short sample, to tell them apart.
And since the ear's pitch detection mechanism is not unlike an analog Fourier transform, a similar thing can happen there.