Pipes partially frozen

[Barbie]

We are experiencing a cold snap... Currently it's -13 C. Because we have a heated hose for water supply the pipes/taps in my bathroom/bedroom are still working however my kitchen keeps freezing. It has frozen twice now but unthawed. I now know to keep the tap open a bit to help stop it from freezing again however in the meantime until it does unthaw, should I have the tap open a bit? Based on forecast we are looking at at least another 24 hours of this cold.