Sounds like the background fabric needs stabilization. I heavily starch the yardage for backgrounds like this before cutting the blocks.
Since you are already working on a block, I would suggest spray starching the back of the block several times (allow to dry between layers and before ironing, to prevent scorching the starch). You can also spray starch the front of the block.
Starching should help a lot. If it still isn't enough, then you need to purchase a stabilizer designed for embroidery to place underneath the background fabric.