Supernovae (SNe) come in various flavors and are classified into different types based on emission and absorption lines in their spectra. SN candidates are now abundant with the advent of large systematic sky surveys like the Zwicky Transient Facility (ZTF), however, the identification bottleneck lies in their spectroscopic confirmation and classification. Fully robotic telescopes with dedicated spectrographs optimized for SN follow-up have eased the burden of data acquisition. However, the task of classifying the spectra still largely rests with the astronomers. Automating this classification step reduces human effort and can make the SN type available sooner to the public. For this purpose, we have developed a deep-learning based program for classifying core-collapse supernovae (CCSNe) with ultra-low resolution spectra from the SED-machine spectrograph on the Palomar 60 inch telescope. The program consists of hierarchical classification task layers, with each layer composed of multiple binary classifiers running in parallel to produce a reliable classification. The binary classifiers utilize recurrent neural networks and convolutional neural networks architecture and are designed to take multiple inputs to supplement spectra with g- and r-band photometry from ZTF. On non-host-contaminated and good quality SEDM spectra ("gold" test set) of CCSNe, CCSNscore is ∼94% accurate in distinguishing between hydrogen-rich (Type II) and hydrogen-poor (Type Ibc) CCSNe. With light curve input, CCSNscore classifies ∼83% of the gold set with high confidence (score ≥0.8 and score-error < 0.05), with ∼98% accuracy. Based on SNIascore's and CCSNscore's real-time performance on bright transients (mpk ≤ 18.5) and our reporting criteria, we expect ∼0.5% (∼4) true SNe Ia to be misclassified as SNe Ibc and ∼6% (∼17) of true CCSNe to be misclassified between Type II and Type Ibc annually on the Transient Name Server.