The phrase "presses on" can be defined as persevering or pushing forward with determination in the face of challenges. There are a variety of synonyms that can be used to convey this meaning, including words such as persists, perseveres, endures, persists, continues, and persists tenaciously. Some other possibilities include terms like soldiers on, persists doggedly, remains steadfast, maintains focus, and sticks with it. These synonyms all emphasize the idea of staying committed and seeing things through to the end, even when the going gets tough. In any situation that calls for perseverance, there are plenty of ways to express the same sentiment as "presses on.