Conditional power (CP) is an important feature frequently implemented in adaptive designs to inform interim decision making such as futility check and sample size re-estimation. This paper adopts the B-value approach and focuses on the cases where the information fraction of the primary endpoint (continuous or binary) is too small (e.g. due to long follow up) to generate reliable CP estimates, but more auxiliary information (e.g. earlier assessment of the primary endpoint) is available. We proposed a framework that not only improves the shift parameter estimate for CP calculation, but also links the auxiliary data with an approximate information fraction without assuming a fixed correlation between the auxiliary data and the primary endpoint. Through the proposed framework, we also provided a benchmark by analytically deriving the true CP that can be used to systematically evaluate various CP estimation methods. The framework also corrects bias in treatment effect estimation due to population shift which further improves CP estimation, if the population shift is captured by the auxiliary data. Compared to existing methods, our proposed framework is more accurate and more robust.