Powering 9VDC from battery
9-volt batteries are made up of six relatively small 1.5V cells which do not deliver much current. As you may know, 9V batteries are often used in small devices that are meant to run for a long time (smoke detector, for example) but draw very little current.

The Energizer 9V (522) battery datasheet shows the battery's milliamp-hour capacity in the following chart:



Along the bottom of the chart you can see various discharge rates from 10 to 50mA. If your device draws 350mA, then it will be far off the chart. At 50mA discharge, the bar in the chart is showing you that the battery will only deliver a little over 400mAh, which equates to about 8 hours (400/50). As the discharge rate goes up, the battery provides less capacity.

Your device 350mA current draw is almost certainly a maximum, but let's assume that it is a constant. Since it is 7 times more than the maximum discharge shown on the chart, it's possible that either the battery cannot supply that amount, or will do so for only a very short time.

I'll assume you just need portability and can't use a line power adapter (wall wart). Your best bet will be to acquire a 6-cell battery holder:



You'll get much more capacity from the larger 1.5V cells. Note also that the model shown has a 9V-style connector.

Since the device probably doesn't draw 350mA all the time, you should be able to get 2 hours with AA size cells. If not, try C size cells.