The 100 m Robert C. Byrd Green Bank Telescope K-band (KFPA) receiver was used to perform a high-sensitivity search for rotational emission lines from complex organic molecules in the cold interstellar medium towards TMC-1 (cyanopolyyne peak), focussing on the identification of new carbon-chain-bearing species as well as molecules of possible prebiotic relevance. We report a detection of the carbon-chain oxide species HC$_7$O and derive a column density of $(7.8pm0.9)times10^{11}$~cm$^{-2}$. This species is theorized to form as a result of associative electron detachment reactions between oxygen atoms and C$_7$H$^-$, and/or reaction of C$_6$H$_2$$^+$ with CO (followed by dissociative electron recombination). Upper limits are given for the related HC$_6$O, C$_6$O and C$_7$O molecules. In addition, we obtained the first detections of emission from individual $^{13}$C isotopologues of HC$_7$N, and derive abundance ratios HC$_7$N/HCCC$^{13}$CCCCN = $110pm16$ and HC$_7$N/HCCCC$^{13}$CCCN = $96pm 11$, indicative of significant $^{13}$C depletion in this species relative to the local interstellar elemental $^{12}$C/$^{13}$C ratio of 60-70. The observed spectral region covered two transitions of HC$_{11}$N, but emission from this species was not detected, and the corresponding column density upper limit is $7.4times10^{10}$ cm$^{-2}$ (at 95% confidence). This is significantly lower than the value of $2.8times10^{11}$ cm$^{-2}$ previously claimed by Bell et al. (1997) and confirms the recent non-detection of HC$_{11}$N in TMC-1 by Loomis et al. (2016). Upper limits were also obtained for the column densities of malononitrile and the nitrogen heterocycles quinoline, isoquinoline and pyrimidine.