Typing woes... 'the convoy ORDER is not cut' not 'the convoy or'

Leonidas is correct.

Example in the case of England fighting a French invasion:
A LVP-WAL (English)
A YOR S F LON H (English) -- CUT BY FRENCH ARMY
F LON S A LVP-WAL (English) -- CUT BY FRENCH FLEET
A WAL-YOR (French)
F ENG-LON (French)
F NTH S F ENG-LON (French)
[assumes that no non-French fleet in BEL, HOL, HEL, DEN, SKA, NWY, NWG, or EDI is ordered to enter NTH]

In the above case, the French fleet in the English Channel would enter London and the English fleet would be dislodged without any available retreat options (NTH, YOR, WAL all occupied and not permitted to retreat to province from which attack was ordered). The French army's order to Yorkshire would fail, but it would be sufficient for the French army to cut Yorkshire's support of London.

Had the French army in Wales instead been ordered to support English Channel to London, THAT support would have been cut by the English order of Liverpool to Wales and the order of English Channel to London would have failed (as the Yorkshire army's support would still be valid). In this second case, Army Wales would not have been dislodged because the support at London would have been cut during the fleet's dislodgement.

Another illustrative case in which Russia is nearly dead against an intruding Austria:
England has a fleet in St. Petersburg.
Russia has armies in Moscow and Warsaw.
Austria has armies in Galicia, Ukraine, and Sevastopol. For whatever reason (probably a threat of a cut support from Armenia or Silesia), it's apparent that Austria should not attack out of Ukraine.
Livonia, Prussia, and are vacant.

Orders:
RUSSIA: A MOS S A WAR; A WAR S A MOS
AUSTRIA: A SEV-MOS, A GAL-WAR, A UKR S A GAL-WAR.
In this case, A SEV-MOS would cut Moscow's support for Warsaw and A GAL-WAR/A UKR S A GAL-WAR would succeed, dislodging Warsaw and leaving Russia vulnerable to losing Moscow in the next season.

Russia could reasonably defend himself against a supported attack on either Warsaw or Moscow by simply ordering:
A WAR-UKR, A MOS-UKR. The result is a self-standoff circumstance in which Austrian support is nullified regardless of which unit Ukraine is ordered to support.

One last case: Russia is invading Turkey. Units are as follows:
RUSSIAN A CON, A ANK
TURKISH A SMY
All neighboring provinces are vacant.

Russia could order A CON-SMY & A ANK S A CON-SMY. Alternately, Russia could order A ANK-SMY & A CON S A ANK-SMY. Turkey cannot order A SMY-CON or A SMY-ANK and in so doing cut support for the attack.

I hope that clears up all non-convoy-related support questions.