This paper proposes a power allocation scheme that improves the overall spectral efficiency (SE) and fairness performance, and reduces a significant amount of per user transmission power for the user-centric cell-free (CF) massive multiple-input multiple-output (mMIMO) systems during uplink transmission. The proposed power allocation scheme comprises of a new power allocation model based on some adjustable parameters and the large-scale fading coefficients, and an algorithm to adapt these parameters for improving the minimum SE performance among all UEs. One important aspect of the proposed scheme is its simplicity from a design perspective which avoids complex optimization methods. Moreover, the proposed scheme provides fairness performance close to that of the max-min-SE based power control strategy and offers a SE performance comparable to that of the power control scheme that maximizes the sum-SE (max-sum-SE) while reducing the average transmission power simultaneously. Numerical results show that, compared to the max-min-SE scheme, max-sum-SE scheme, and a recent fractional power allocation policy, the proposed power allocation scheme improves the SE performance by up to 64.3% while reducing the average transmission power by up to 89%.