Organic matter in regard to gardening is anything that can decompose. Kitchen scraps, leaves, pine needles, yard trimmings, contents of mower bag, pulled weeds, mulch, shredded paper, chipped wood, compost. Compost is what organic matter is called when it has decomposed enough to be unrecognizable. Larger pieces of wood & bone will also decompose, but do so at such a slow rate that they are not usually included in composting unless one has space to devote to a long-term pile.
Please know that clay + sand = concrete. Continually adding organic matter really is all that is needed to turn any kind of soil into "black gold" fertile garden soil. There is no need to do anything with it but put it on the soil surface, the way mother nature gardens when no humans are around to intervene. Over time, the soil will become more fertility, and will drain better in times of much rain, while not drying as quickly during times of less.
I've seen this work when I lived in OH in housing developments where the top soil is sold before the neighborhood is built, leaving nothing but the lower layer of solid clay, and in AL where the soil is so sandy that soil is always dusty dry the day after a hard rain.
But don't take my word for it, Dr. Elaine Ingham, soil microbiologist, can explain it in 15 minutes: