the best way to add/modify the plot data is by XData, YData, ZData, or XDataSource, ... properties. Probably, by changing their properties, callback to automatically update the plot will be executed and there won't be need for refresh button.
In AppDesigner, you need to provide the handle to the UIAxes component you're plotting on, then it won't open the new figure.
x = 1;
y = 1;
h = plot(app.UIAxes, x, y);
If you want to change the data (this is useful if you have a lot of points in your graph), here's an example
xNew = [x, 2];
yNew = [y, 3];
set(h, 'XData', xNew, 'YData', yNew);
if not, then you can use the plot function like shown above.