Glass-cell-based secondary clocks, including coherent population trapping (CPT) clocks, are the most used clocks in modern laboratories and in industry. However, the reported frequency accuracies of those secondary clocks were always much worse than expected, though all error sources have been previously discussed. In this report, a high-precision measurement on the spectral frequency-linewidth relation (FL-R) is first used for revealing a new error source in secondary clocks by which we answer the puzzle raised in Opt. Lett.38, 3186 (2013)10.1364/OL.38.003186.