Both 漸 (ぜん) and 次 (じ) are standard on'yomi. 漸 means gradually or progressively. Wrong readings like せんじ mispronounce 漸, and ぜんつぎ uses a kun'yomi for 次. It means gradually, little by little, or progressively over time, indicating a slow and steady process.

漸 (ぜん) も 次 (じ) も標準的な音読み。少しずつ、次第に物事が進む様子を表します。段階的に、ゆっくりと変化していくことです。