Besought (parekaloun). Imperfect active, began and kept on beseeching. This is the same verb used by Matthew in Mat 8:5 of the centurion himself.
Earnestly (spoudaiōs). From spoudē haste. So eagerly, earnestly, zealously, for time was short.
That thou shouldst do this for him (hōi parexēi touto). Second future middle singular of parechō. Old and common verb, furnish on thy part. Hōi is relative in dative case almost with notion of contemplated result (Robertson, Grammar, p. 961).