It's likely that this note cannot be played cleanly on your recorder.
There are two fingering systems that may be found on modern recorders:
- English (Baroque) - a reconstruction of the baroque recorder, based in large part on the work of Arnold Dolmetsch. This system can be identified by the 4th hole (counting from the lip) being smaller than the 5th hole.
- German ("school") - Peter Harlan's simplified fingering, where the 5th hole is smaller than the 4th hole. This allows the ascending major scale to be played simply by consecutively lifting fingers - at the cost of making chromatic notes harder to play in tune.
Judging by your description, your instrument is the lowest tier of school recorder in German fingering, meaning that it is made to play cleanly in C major only. This is just about sufficient to play the kind of basic pieces one might encounter during general music lessons in school, but obviously insufficient for any kind of advanced performance.
You could possibly try to achieve a clean note through shading (partial covering of holes), breath control, or possibly even an alternative fingering - but it may be a better idea to get a quality baroque system recorder, if you wish to pursue the instrument more seriously.
I'm afraid there's nothing I can tell you about the make of your recorder.
Addendum to address the added image:
As I had guessed, the instrument in question is a very basic school recorder with German fingering. Trying to perform advanced material on it is likely to be an exercise in futility.
Buying recommendations are off-topic here, but any baroque-fingering recorder from a reputable manufacturer should be adequate.