It is a small but good step. You need a good mulch. Powdered sulfur is a much better acidifier if your plant is chlorotic (yellowing of the leaves with green veins). Before applying the sulfur, pull the mulch back and then sprinkle the mulch back over the soil after sprinkling the powdered sulfur directly on the soil. On my website below is a chart of how much sulfur to apply. Do not cultivate around your Cunninham's White. It has very shallow roots and cultivating will severely harm the roots. But do mulch. Mulch protects the roots, keeps the soil moisture stable, and keeps the roots cool.

Long-term use of the same type of mulch may change the surface soil's pH. Acidic mulches such as pine bark and pine needles may have a pH of 3.5 to 4.5. When you apply them continually, they may, over several years, cause the surface soil to become acidic. Acidic mulches such as pine needles are beneficial for maintaining the acidic conditions required by acid lovers such as rhododendrons and azaleas.

Conversely, hardwood bark mulch (even though it is initially acidic) may cause the surface soil to eventually become too alkaline (soil pH above 7.0), causing shallow-rooted, acid-loving plants to decline because of micronutrient deficiencies (the chief problem stemming from high pH levels).

Studies have shown that mulches do not significantly alter subsurface soil pH because of the buffering capacity of the soil. Therefore, concerns about pH are mostly relevant to shallow-rooted plants such as rhododendrons.

Here is a link that might be useful: How to grow rhododendrons and azaleas.