Derry were totally outclassed in their Division 1B match against Cork at Pairc Ui Chaoimh on Sunday.
They trailed the Rebel County by 17 points at the break having only managed to score three points in the first half.
In contrast, Cork scored at will and had registered a massive 2-14 tally at the end of the first period.
Lavey's Enda Cushnahan popped up to give Derry their only goal of the game while their points came from Ollie Collins (2) and Paul Curran, Sean McGlone and Raymond Kennedy.
Derry have lost all three of their league outings this season and face a tough challenge against Tipperary next weekend.
In Division 2A, Down now have two wins from three games after a convinvicing win over Kildare at Portaferry.
Allianz National Hurling League
Division 1A
Waterford 2-11 Dublin 2-13
Clare 2-16 Galway 1-18
Laois 1-08 Kilkenny 1-17
Division 1B
Cork 6-19 Derry 1-05
Tipperary 3-21 Wexford 1-11
Limerick 1-12 Offaly 2-16
Division 2A
Down 2-22 Kildare 1-07
Roscommon 0-11 Westmeath 3-12
Division 2B
Meath 0-11 Kerry 2-19
Carlow 2-19 Longford 1-09
Division 3A
Leitrim 1-09 Donegal 2-09
Sligo 3-10 Mayo 2-11 (played Saturday)
Division 3B
Fermanagh 2-04 Monaghan 3-12
Cavan 0-06 Tyrone 2-17