What is the authenticity of the saying, "A morsel in the belly of a hungry person is better than building a thousand mosques," attributed to Sheikh Abdul Qadir Al-Jilani, and is it permissible to propagate it?

This statement is not attributed to Shaykh Abd al-Qadir al-Jilani by reliable authorities. The first part of it, "A morsel in the belly of a hungry person is better than building a thousand mosques," is not a hadith (definition), and it carries its own falsehood within it, because the reward for a morsel in the belly of a hungry person cannot possibly be greater than building a thousand mosques. The rewards for deeds and the preference between them are based on transmission (naql), not on assertion without proof.