We present medium resolution near-infrared spectral measurements of the carbon monoxide (CO) and the cyano radical (CN) features in 12 Galactic classical Cepheids. The pulsation periods of our sample range from 5.5 to 69 days, and the stars studied each had five or more near-IR spectral observations. The CO and CN measurements were used to probe CNO abundances of these stars, and elemental abundance values from the literature were used to identify the trends of [C/N] and [O/N] with CN and CO. To put these measurements in context, we performed stellar atmosphere fitting to obtain estimates of stellar parameters, with a primary focus on effective temperature. Our measurements and temperature estimates show that CN is significantly affected by dredge-up of processed material. We provide discussion as to the potential nature of the recently confirmed classical Cepheid, ET Vul, and connect our near-infrared CO measurements to the mid-infrared period-colour-metallicity relation.