Google Sheet App Script function that will trigger an refresh of cells that are dependent on it.

trigger_refresh.js

//
// Trigger function that updates just one cell 'MySheet:$A$1`.
//
// Why? So that it triggers a refresh!
//
// How? Thats the fun part, if you add dependent cells to a 
//      user formula it causes a refresh.
//
// You can tell the spreadsheet which cells the function is 
// dependent on by passing them to the formula at the end of  
// its expected parameters. 
//
// For Example:
//
//   =MyFunction("this", "that", MySheet:$A$1)
//
// This tells the sheet that it is dependent on MySheet:$A$1 and to 
// refresh if that cell changes.
//
// You have to create a Trigger to make this work.
//
function triggerRefresh() {
  let sheet = SpreadsheetApp.getActiveSpreadsheet().getSheetByName('Current')
  let cell = sheet.getRange("$A$1");

  // Make a "change" that will cause cells depend on it recalculate
  var currentValue = cell.getValue();
  cell.setValue(currentValue); 

  // Make a comment on the cell what time data was refreshed
  const now = new Date(); 
  cell.setComment(`Updated data at ${now}`);
  console.log(`Updated data at ${now}`)
}

To create a trigger in Google Sheets using Apps Script, you can use the script editor to manually create a trigger.

Manual creation

• Open the Apps Script project
• Click the Triggers icon, which looks like a clock
• Click Add Trigger
• Choose the function to trigger
• Select the trigger type, such as time-driven or event-driven
• Configure the trigger settings
• Click Save