Complex sentence: Around this time she married guitarist and singer Ernest Lawlars (a.k.a. Little Son Joe) and began recording material with him in 1939, with Son's playing adding a more rhythmic backing to Minnies's guitar.
Split into simpler sentences: Around this time she married the guitarist and singer Ernest Lawlars, known as Little Son Joe. They began recording together in 1939, with Son's playing adding a more rhythmic backing to Minnies's guitar.