The phrase "beg indulgence" is often used to politely request permission or forgiveness from someone in a formal setting. However, there are many other synonyms that can be used in its place, depending on the specific context and tone of the conversation. These might include phrases such as "seeking your understanding," "asking for your pardon," or "pleading for your leniency." Other options might include phrases like "requesting your patience," "seeking your forgiveness," or "asking for your forbearance." Ultimately, the choice of synonym will depend on the specific situation and the level of formality or informality being conveyed in the conversation or written communication.