There are several ways to deal with horse archers, but most involve strategy and tactics. Hoping a single super weapon will deal with the enemy is a fools game (even super weapons need a proper strategy to be effective).
To provide an illustration, the Battle of Nagashino will serve as a substitute. Tokugawa Ieyasu and Oda Nobunaga combined forces against the army of Takeda Katsuyoui, who had inherited a highly effective cavalry force from his father, Takeda Shingen.
The combined force had access to firearms, which can be substituted for the crossbow weapons, since the effect is going to be the same. By placing the ranks of arquebus gunners (crossbow archers) behind wooden palisades and having them fire in volleys, with fresh troops replacing each line of gunners (archers) as they fired, the cavalry charge was repelled and the formation of the combined army was unbroken when the Takeda infantry force arrived behind the cavalry charge.
Of course, the topography also assisted Tokugawa Ieyasu and Oda Nobunaga's force, since the flanks were protected by ground the cavalry could not use to manoeuvre around, and had they not chosen this place to make their stand, the use of volley fire might have been far less effective.
Henry V made a similar calculation at Agincourt, securing his flanks with heavy woods, and finding a place where the French had to charge uphill against the ranks of Welsh longbowmen and dismounted men at arms. If caught in the open (as English forces were later in the 100 Years War), the dismounted archers were easily overwhelmed.
So the use of these crossbows will make your soldiers marginally more effective in a battle (they can engage at longer ranges and possibly punch through heavier armour, although this isn't a huge factor against the Huns), but the mobility and firepower of the horse archers is what makes them effective, so the generals need to find ways to negate those factors, or match them with better mobility and firepower of their own.