MOVES & Co does not register Artists for performance or neighbouring rights directly, you can register by searching for your local neighbouring-rights/performers' society collection society
You may be entitled to royalties whenever your music is played on radio, TV, or in public, separate from streaming, through your country's local rights collection society (eg PPL (UK), MCSN (Nigeria), GHAMRO (Ghana), UPRS (Uganda), SAMPRA (South Africa), COSOTA (Tanzania), BURIDA (Côte d'Ivoire), SODAV (Senegal).
Performance rights: is money for the song itself (the lyrics and melody). Whoever wrote it earns this every time it's played publicly, on radio, streaming, TV, or live. This is what PRS (UK) and similar bodies collect.
Neighbouring rights: is money for the actual recording. Whoever performed on it, or owns the master (artist, session musicians, label), earns this every time that specific recording is played publicly. This is what PPL (UK) and similar bodies collect.
If someone covers your song, you still get paid as the songwriter (performance rights) but you don't get paid for their recording of it (neighbouring rights), because that's a different performance and a different master.