Probably cognate with Mandarin 冬菇 (dōnggū, “winter mushroom”). The final 子 appears to be ateji for 菇, as the latter is not used very much in Japanese.
冬子 (hiragana どんこ, romaji donko)
- a type of shiitake mushroom grown during the winter
- Winter cultivation produces a mushroom with thicker flesh, regarded as a higher-quality dried product.