In 2007, Andrews introduced the k-marked odd Durfee symbols and studied their parities. Let Dk0(n) denote the number of k-marked odd Durfee symbols of n. He proved that Dk0(n) is even for k≥1 and n≡k−1(mod2). Inspired by Andrews' work, many authors investigated congruences for Dk0(n) modulo 2 or powers of 2. For example, Wang obtained D20(8n+4)≡D20(8n+6)≡0(mod2) which were first conjectured by Andrews. More recently, Xia proved that D2mk0(2n+1)≡0(mod2m+1) with m,k≥1 and n≥0. In this paper, we first study the modular properties of the generating function of D20(n) which is completed to be a weakly modular form of weight 52. As application, we utilize the Hecke operators to establish families of congruences for D20(n) modulo powers of primes ℓ≥5.