Here are some synonyms for imploring:
* Begging
* Pleading
* Entreating
* Supplicating
* Praying
Examples:
* The child implored her mother to buy her a new toy.
* The prisoner implored the judge for mercy.
* The refugees implored the authorities to let them enter the country.
Imploring is often used in situations where someone is facing a difficult situation or is in need of something essential. It can also be used to describe a heartfelt request, even if the need is not desperate.