Yes according to the Rangers Companion feature of the Beast Master archetype, the companion is trained and works in partnership with the Ranger.
The beast obeys your commands as best as it can. (PHB p93)
Since the carry rules apply to any character including a beast companion, the beast can carry. Therefor the bloodhawk will obey a beast master’s command to carry them.
This depends on the beast being strong enough to carry the PC. We can look at the carrying rules in more detail.
Your carrying capacity is your Strength score
multiplied by 15 [in pounds]. (PHB p176)
Bloodhawk has a strength of 6, so can carry 90 pounds.
Using the Variant Encumbrance rules, carry weight is 5 times Strength score before being encumbered. Carry in excess of 10 times Strength and become heavily encumbered.
With the variant rule, the bloodhawk can carry 30lb’s before suffering a speed penalty of 10 feet. Carry more than 60lb’s and speed drops 20 feet, suffer disadvantage on ability checks, attack rolls, and saving throws that use Str, Dex, or Con.