Weave broken. Coat broken ends of straw and body of weave with a little Elmer's Glue-All. Cover glued surface with wax paper. Then clamp the break between boards. If necessary, break can be reinforced on the wrong side of the straw with cloth coated with Elmer's Glue-All.
Holes. Leave ends of broken straws jagged: they will be less obvious than if you cut them off cleanly. Cut a patch out of matching straw, coat edges with Elmer's Glue-AU and paste under the hole. Cover both sides of straw with wax paper and clamp patch between boards.
A better-looking mend
is made by weaving in new straw, but this is possible only on simple, coarse weaves like those used in old farm hats.