Let the variegated ash blossom in the spring before pruning. The exact time of blooming depends on your hardiness zone, with blooms coming in as early as March in hardiness zone 9 and as late as May in zone 3. Buds form the season before they bloom, so pruning at other times of the year removes buds and reduces the number of blossoms.
Trim back the blossoms with pruning shears as soon as the blossoms have died out. This allows the shrub to expend energy on new blossoms and plant growth, helping the slow-growing shrub to build up its size.
Cut back any undesired growth with pruning shears after the blossoms have died out but before the leaves have fully emerged. Trim back the sides of the bush to encourage upward growth and trim the top to encourage the plant to fill out.
Cut off any dead branches with pruning shears, as soon as you notice them, at any point in the year.