The reasons why we didn't unclude native-Bulgarian horse archers and have low levels of cavalry are many, but I can simply summarize it to: the old Bulgars were semi-nomadic people, yes, but they did mix with the Slavs and after the Russian attack (around 969) it seems we've lost much of our Bulgar equestrian nobility (at least in the wars between 970 and 1018 we have almost no cavalry). After 1018 Bulgaria fell under Byzantine rule, which lasted till 1185 and imposed a really huge influence (which was pretty big after the Christianization anyway), so after the Second Bulgarian state was restored in 1185, it mostly copied the Byzantine system in a number of ways, including armament and style of warfare. The light cavalry and the horse archers were almost entirely provided by mercenaries and allies - most usually Cumans (who were Bulgarian allies till the Mongol invasion) and after the Mongols came, some Cumans settled in Bulgaria, while later on we started using Tatar mercenary horse archers (also Alans some times).
So, basically, the reasons for the lack of horse archers and light cavalry are because the Bulgarian people changed a lot during those times, especially in the socio-economical style of living (good steppe horse archers require a steppe way of living, while later Bulgarians became more and more feudalized etc), so during the SBE the Bulgarians were too Byzantinized to field their own horse archer armies. Fortunately for them, they were best buddies with the Cumans, who filled that niche perfectly...
As for the Pechenegs, there might have been some remnants of them by 1190, but they would probably be too minimal to make separate military detachments (the only sizeable and possibly non-Bulgarian units were the Cumans and the Vlachs) - the Pechenegs (and the Uzes before them) got assimilated, Christianized and settled down rather fast (unlike the Cumans, who kept their way of living and self-consciousness in Bulgaria till around the Turkish times, although they did convert to Christianity).