Government figures at:
http://www.dti.gov.uk/energy/bluebook/pdf/annex3.pdf
show that there was a sweeping reduction in hydrocarbon taxes on the
8th of August 1977. However, I can find no information on why this
reduction was passed. While the duty levels had been intermittantly
frozen, there had never been an explicit cut before this time.

I would like to know more about this cut and what political pressures
caused it to be passed.

For a complete view of the "political pressures", you may need to
visit a library which keeps back issues of "The Times" on microfilm.

These bits and pieces suggest you focus on late March, April and July.

"March 30 1977
PETROL today went up by 5½p per gallon following Chancellor Healey's
Budget. A survey on new prices facing Bolton motorists showed that
drivers can expect to pay between 79½p to 83½p for a gallon of four
star."
http://www.thisisbolton.co.uk/lancashire/bolton/bygone/MAR02BYGONE33.html

"April 3, 1977
THE Liberals will push for an amendment to the Finance Bill to axe the
new petrol tax, introduced in last week's Budget. They hoped the move
would take the 5½p a gallon off petrol by the summer."
http://www.thisisbolton.co.uk/lancashire/bolton/bygone/APR02BYGONE31.html

"1977 Budget (Denis Healey)
[...]
A proposed increase in petrol duty was dropped after Liberal pressure
in Parliament (despite the recent Lib-Lab pact)."
http://www.bbc.co.uk/budget96/background/Budgets-1945-1979.htm

Budget day was 29 March 1977, with a "mini-budget" on 15 July. 
http://www.bized.ac.uk/dataserv/chron/kf5579.htm#1977

"Attack on inflation after 31st July 1977
Corporate author:	Chancellor of the Exchequer
Abstract:	A White paper which outlines further measures which the
Government proposes to take against inflation and unemployment [...]
On tax relief the author presents a recommendation to reduce petrol
duty, but no reductions are recommended in other areas."
http://www.bopcris.ac.uk/bop1974/ref2641.html

Philip Larkin wrote a letter to "Robert Conquest (1 April 1977) in
which he reacts to a government budget discussion by saying that, "The
Budget put a bob on petrol, which caused a frightful squeal, so now
the Liberals are going to get it put on drink. As a drinking motorist
this doesn't make much difference to me, but the price of drink is
really fierce. "
http://www.northallertoncoll.org.uk/english/alit/Larkin/Antigon.htm