It can be done this way.

    location /my_app {
        rewrite ^/also/?(.*)$ /$1 break;
        proxy_pass http://my_app_upstream;
        proxy_set_header Host $host;
        proxy_set_header X-Real-IP $remote_addr;
        proxy_set_header X-Forwarded-For $proxy_add_x_forwarded_for;
        ...
    }

Without rewrite it will pass /my_app to upstream, so prefix should be handled on my_app level.

    location /my_app {
        proxy_pass http://my_app_upstream;

And with / in upstream it will add an additional /, so urls in proxied service will look like //some-url

    location /my_app {
        proxy_pass http://my_app_upstream/;