The phrase "with one purpose" refers to having a clear, singular goal or objective in mind. There are many synonyms for this phrase that can be used to convey the same meaning. One such synonym is "single-mindedly," which emphasizes the unwavering focus on one particular aim. Another option is "determinedly," which implies a steadfast resolve to reach a specific outcome. "Singularly" is another synonym that underscores the idea of having just one intention or objective. It's important to choose the appropriate synonym based on the context of the sentence and the desired emphasis on the singular purpose.