It is widely suspected that the anaerobic bacterial species Propionibacterium acnes ( P. acnes ) contributes to the development of acne, but its exact role is not well understood.  There are specific sub-strains of P. acnes associated with normal skin, and moderate or severe inflammatory acne.  It is unclear whether these undesirable strains evolve on-site or are acquired, or possibly both depending on the person. These strains have the capability of changing, perpetuating, or adapting to the abnormal cycle of inflammation, oil production, and inadequate sloughing of dead skin cells from acne pores. Infection with the parasitic mite Demodex is associated with the development of acne.   It is unclear whether eradication of the mite improves acne. 
Topical hydrocortisone works much higher up in the skin. It is not a treatment for acne or acne cysts. In general, this category of products, especially its higher potency topicals, can carry side effect of skin thinning. As far as an acne cyst is concerned, it lies much deeper in the skin, and it is unlikely that a weak topical such as OTC hydrocortisone can reach it to have any significant improvement. The skin overlying the cyst surface is already stretched or thin to begin with; therefore, introducing any topical cortisone for its treatment can risk irreversible skin thinning. Another important side effect of cortisone to be taken into consideration here is "rebound or worsening of acne." Yes, these products can contribute to more acne.