Difference between one-way and two-way ANOVA

Let me preface my question with a summary of my understanding of ANOVA.

1. One-way ANOVA: When the researcher is interested in the effect of one independent variable (such as treatment) on a dependent variable. There may be other variables that affect the dep. variable, but that effect is not removed in any way in one-way ANOVA.
2. Two-way ANOVA: When the researcher is interested in the effect of two independent variables on the dep. variable. The P-values that you get from a two-way ANOVA have mathematically taken into consideration the other independent variable.

My question: Since we are told that if we are only interested in the effect of 1 independent variable we should use a one-way ANOVA, is there any way to remove variability of a 2nd ind. variable? Could we perhaps do a two-way ANOVA with that 2nd variable even though we aren't interested in how it affects the dep. variable. We would simply want to remove the variability of the 2nd ind. variable from the P-value of our 1st ind. variable.

Notes: Please let me know if my understand of ANOVA is flawed. I have looked many places for clarification on this question and have struggled with it for years.

• What is "to remove" for you? Either "ignore" or "take into account to control for"? Commented May 19 at 12:03