What effect does verbal and mental abuse have on a child later in life?

The death of trust. A kids self concept is relatively set by the age of 6-8. The parent or parents provide the most important trust relationship in that formative period. If the parent (who they idolize by default) tells them they are worthless or variations of the same, the ability to trust can die & self concept is wounded. Many adult problems or addictions reflect a desire to fill the void left by this wound.
See below. There can be multiple problems. The child may have trouble with anxiety, low self esteem, depression, conduct disorder, and/or op-positional defiant disorder. The child may grow up thinking these are normal behaviors, and pattern those behaviors as an adult.