After eating non-vegetarian food(like, chicken), if I only wash my hands and do not take bath; then is it permissible for me to enter the mosque and pray?
There's a hadith addressing this:
Narrated Ibn 'Abbas: The Prophet (ﷺ) ate of the meat of a shoulder (by cutting the meat with his teeth), and then got up and offered the prayer without performing the ablution anew. -- Sahih al-Bukhari 5404 (sunnah.com)
(See also Sunan Abi Dawud 187 (sunnah.com).) So the Prophet made prayer after eating non-vegetarian food.
However, wudu is needed after camel meat; see Does eating camel's meat break the wudu?.
As far as i know only when ine eats camel meat should renew their Waddu.