I am using nginx and want to expose Firefly III under /budget/

The following snippet might help:

location ^~ /firefly-iii/ {
   deny all;
}

location ^~ /budget {
   alias /var/www/firefly-iii/public;
   try_files $uri $uri/ @budget;

   location ~* \.php(?:$|/) {
      include snippets/fastcgi-php.conf;
      fastcgi_param SCRIPT_FILENAME $request_filename;
      fastcgi_param modHeadersAvailable true; #Avoid sending the security headers twice
      fastcgi_pass unix:/run/php/php7.4-fpm.sock;
   }
}

location @budget {
   rewrite ^/budget/(.*)$ /budget/index.php/$1 last;
}