Comparing a clean, noisy and enhanced signal
The answer to you question depends on how quality is defined in your specific application. Since you're working with audio signals, it is often beneficial to use the perceived quality to analyze the efficiency of an audio enhancement algorithm.
How to Determine the Perceived Quality
Methods to determine the perceived quality of an audio signal can be categorized into

subjective and
objective

methods. Subjective methods are based on human listeners that rate the audio signal according to some defined criteria. Objective methods use algorithms that use some measure, modeling parts of the human auditory system, to estimate the perceived quality.
Listening Tests
Usually, the most accurate way to determine the perceived quality is to perform a listening test and ask human listeners to rate the quality of different signals. Listening test methods comprise, e.g., the MUltiple Stimuli with Hidden Reference and Anchor and pairwise comparison approaches.
Objective Measures
A simple objective measure surely is the signal-to-noise ratio (SNR). In many cases, though, the SNR is only related to the actual perceived quality to a certain extent. Therefore, more involved objective measures have been created. Well-known ones are

the Perceptual Evaluation of Speech Quality (for speech signals) and
Perceptual Evaluation of Audio Quality (for audio/music signals)

algorithms.