I have an original language copy of The Arrivall of Edward IV. What is clear is that, for the 1471 campaign at least, Edward was served by some very good light/medium cavalry. We are talking at least hobblers, scourers or custrells. Unarmored horses, half-armour and helmets on the riders, at least a boar spear or long sword and maybe a true lance. Also called 'prickers' or 'spears'.
For example when Edward headed for London, from the Midlands, The Arrivall notes that his 'behind riders' (rearguard) successfully held off any of the enemy's aforeriders from attacking his army on the road.
Later in the same campaign, his afore-riders (scouts) detected the Lancastrian feint towards Wiltshire and London when the Lancastrians were actually heading for South Wales and the Severn river crossings. This enabled Edward to corner their army at Tewkesbury. And, of course, it was at Tewkesbury that Edward's 200 horse famously flanked Somerset's unit by charging out of an ambush position in a wooded park. This is one of the few reliably documented use of cavalry in England in this period.
With regard to Barnet we can assume that the same light horse were with Edward and were PROBABLY deployed as rear supports behind his army in the early stages – a bit like field police or provost guard – to stop any deserters from slipping away from the battlefield.
The fog at Barnet may have prevented them from seeing Hasting's unit rout and may have prevented them from thus hitting Oxford's unit in the flank as some of Oxford's men pursued Hastings to the rear. It all depends on how thick the fog really was.
If the fog lifted after the Oxford/Neville friendly fire incident then Edward might have released his light horse to pursue the routing Lancastrian army but if the fog remained down then (if I was Edward) I would have kept the light horse 'in hand' in case of nasty surprises. He was an astute lad so use your own judgement.
Short answer: "assume at least 200 good and well-motivated light/medium cavalry
with Edward at Barnet."
Barry