The C-Band All-Sky Survey C-BASS is a high-sensitivity all-sky radio survey at an angular resolution of 45 arcmin and a frequency of 4.7 GHz. We present a total intensity 4.7 GHz map of the North Celestial Pole (NCP) region of sky, above declination +80 deg, which is limited by source confusion at a level of ~0.6 mK rms. We apply the template-fitting (cross-correlation) technique to WMAP and Planck data, using the C-BASS map as the synchrotron template, to investigate the contribution of diffuse foreground emission at frequencies ~20-40 GHz. We quantify the anomalous microwave emission (AME) that is correlated with far-infrared dust emission. The AME amplitude does not change significantly (<10%) when using the higher frequency C-BASS 4.7 GHz template instead of the traditional Haslam 408 MHz map as a tracer of synchrotron radiation. We measure template coefficients of $9.93pm0.35$ and $9.52pm0.34$ K per unit $tau_{353}$ when using the Haslam and C-BASS synchrotron templates, respectively. The AME contributes $55pm2,mu$K rms at 22.8 GHz and accounts for ~60% of the total foreground emission. Our results suggest that a harder (flatter spectrum) component of synchrotron emission is not dominant at frequencies >5 GHz; the best-fitting synchrotron temperature spectral index is $beta=-2.91pm0.04$ from 4.7 to 22.8 GHz and $beta=-2.85pm0.14$ from 22.8 to 44.1 GHz. Free-free emission is weak, contributing ~$7,mu$K rms (~7%) at 22.8 GHz. The best explanation for the AME is still electric dipole emission from small spinning dust grains.
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