The phrase "much prodding" means that someone has been pushed or encouraged repeatedly in order to get them to do something or provide information. Full definition
The Farewell Party (Israel) Israel's Oscars, «the Ophirs,» have heartily embraced this dramedy about senior citizens who accidentally and after much prodding become crusaders for mercy killings of their terminally sick friends and spouses. (thefilmexperience.net)
The fighters did not need much prodding either. (si.com)
Still, it didn't take much prodding to get dozens of people of all generations to email me and let me know what speaking habits they think Millennials need to stop using now — at least at work. (inc.com)