Please see link above, sorry about the sound/quality you’ll have to turn it up a little…
My 2000 1.6LSi (D16Y7) Civic Coupe EJ6 does this strange “whirring” noise when you cold start it – only ever first thing on a morning or when I finish work and it’s had a good 7/8 hours in the shade. You notice the tone changes when I turn the wheel slightly? It soon fades and only ever lasts about a minute.
It’s been recently serviced only 2 months ago and all fluid levels seem fine.
I just realized this: It sounds like your power steering belt is too tight. When a belt is too tight, it will make a whirring noise. Either that or it could be a bad bearing on the power steering pump pulley.
It is pretty straightforward to loosen or tighten the PS belt. You could easily do the job yourself if so motivated. Replacing the PS pump is expensive but also not very difficult to do.