GFF is a tab-separated format but you are are not using tabs. Unless you use -F'\t' or BEGIN{FS="\t"}, awk will use any whitespace as the field delimiter, and that includes spaces. Since you are cutting on spaces, $9 ends at the first space. You also don't need two commands, what you want to do is:
$ awk -F'\t' '$3=="mRNA"{print $9}' file.gff
ID=Nitab4.5_0006317g0010.1;Parent=Nitab4.5_0006317g0010;Name=Nitab4.5_0006317g0010.1;_AED=0.08;_eAED=0.08;_QI=0|0.45|0.25|1|0.90|0.75|12|0|1011;Note="Peptidase S59%2C nucleoporin"
ID=Nitab4.5_0006317g0020.1;Parent=Nitab4.5_0006317g0020;Name=Nitab4.5_0006317g0020.1;_AED=0.26;_eAED=0.26;_QI=15|0|0|0.83|0.6|0.33|6|0|424;Note="Putative S-adenosyl-L-methionine-dependent methyltransferase"
ID=Nitab4.5_0006317g0030.1;Parent=Nitab4.5_0006317g0030;Name=Nitab4.5_0006317g0030.1;_AED=0.01;_eAED=0.01;_QI=161|1|1|1|0|0.5|2|358|141;Note="Unknown"
ID=Nitab4.5_0006317g0040.1;Parent=Nitab4.5_0006317g0040;Name=Nitab4.5_0006317g0040.1;_AED=0.02;_eAED=0.02;_QI=0|0|0|1|1|1|3|0|187;Note="Heavy metal-associated domain%2C HMA"
ID=Nitab4.5_0006317g0050.1;Parent=Nitab4.5_0006317g0050;Name=Nitab4.5_0006317g0050.1;_AED=0.24;_eAED=0.24;_QI=0|0|0|0.5|1|1|2|0|72;Note="Unknown"
ID=Nitab4.5_0002367g0010.1;Parent=Nitab4.5_0002367g0010;Name=Nitab4.5_0002367g0010.1;_AED=0.11;_eAED=0.11;_QI=0|0|0|1|1|1|14|0|668;Note="Auxin response factor%2C B3 DNA binding domain%2C DNA-binding pseudobarrel domain%2C AUX/IAA protein%2C Aux/IAA-ARF-dimerisation"
ID=Nitab4.5_0002367g0020.1;Parent=Nitab4.5_0002367g0020;Name=Nitab4.5_0002367g0020.1;_AED=0.26;_eAED=0.26;_QI=0|0.5|0|0.66|0|0|3|0|258;Note="Cellulose synthase"
ID=Nitab4.5_0002367g0030.1;Parent=Nitab4.5_0002367g0030;Name=Nitab4.5_0002367g0030.1;_AED=0.33;_eAED=0.33;_QI=0|0|0|0.5|0|0.5|2|0|213;Note="Cellulose synthase"
ID=Nitab4.5_0002367g0040.1;Parent=Nitab4.5_0002367g0040;Name=Nitab4.5_0002367g0040.1;_AED=0.09;_eAED=0.09;_QI=0|0|0|1|0.5|0.66|3|0|230;Note="Zinc finger%2C RING-type%2C Zinc finger%2C RING/FYVE/PHD-type"
ID=Nitab4.5_0002367g0050.1;Parent=Nitab4.5_0002367g0050;Name=Nitab4.5_0002367g0050.1;_AED=0.08;_eAED=0.08;_QI=75|1|1|1|1|1|6|146|267;Note="SAC3/GANP/Nin1/mts3/eIF-3 p25%2C 26S proteasome non-ATPase regulatory subunit Rpn12"
If you only want to get the value of Note=, you can do:
$ awk -F"\t" '$3=="mRNA"{sub(/.*Note=/,"",$9); print $9}' file.gff
"Peptidase S59%2C nucleoporin"
"Putative S-adenosyl-L-methionine-dependent methyltransferase"
"Unknown"
"Heavy metal-associated domain%2C HMA"
"Unknown"
"Auxin response factor%2C B3 DNA binding domain%2C DNA-binding pseudobarrel domain%2C AUX/IAA protein%2C Aux/IAA-ARF-dimerisation"
"Cellulose synthase"
"Cellulose synthase"
"Zinc finger%2C RING-type%2C Zinc finger%2C RING/FYVE/PHD-type"
"SAC3/GANP/Nin1/mts3/eIF-3 p25%2C 26S proteasome non-ATPase regulatory subunit Rpn12"
And, to remove the quotes at the beginning and end of the Note, while keeping any that might be part of note itself:
$ awk -F"\t" '$3=="mRNA"{sub(/.*Note=/,"",$9); print $9}' file.gff | sed 's/^"//; s/"$//'
Peptidase S59%2C nucleoporin
Putative S-adenosyl-L-methionine-dependent methyltransferase
Unknown
Heavy metal-associated domain%2C HMA
Unknown
Auxin response factor%2C B3 DNA binding domain%2C DNA-binding pseudobarrel domain%2C AUX/IAA protein%2C Aux/IAA-ARF-dimerisation
Cellulose synthase
Cellulose synthase
Zinc finger%2C RING-type%2C Zinc finger%2C RING/FYVE/PHD-type
SAC3/GANP/Nin1/mts3/eIF-3 p25%2C 26S proteasome non-ATPase regulatory subunit Rpn12
You could also do the whole thing with GNU grep and some regex magic:
$ grep -oP '\tmRNA\t.*Note="\K[^"]+' file.gff
Peptidase S59%2C nucleoporin
Putative S-adenosyl-L-methionine-dependent methyltransferase
Unknown
Heavy metal-associated domain%2C HMA
Unknown
Auxin response factor%2C B3 DNA binding domain%2C DNA-binding pseudobarrel domain%2C AUX/IAA protein%2C Aux/IAA-ARF-dimerisation
Cellulose synthase
Cellulose synthase
Zinc finger%2C RING-type%2C Zinc finger%2C RING/FYVE/PHD-type
SAC3/GANP/Nin1/mts3/eIF-3 p25%2C 26S proteasome non-ATPase regulatory subunit Rpn12
The -o makes grep print only the matching portion of the line, and the -P enables Perl Compatible Regular Expressions (PCRE). The \K, a PCRE feature, means "discard anything matched before this point". So, the regex looks for mRNA surrounded by tabs (\tmRNA\t; note that this means it will match if mRNA is the entire value of any field, but that should never happen in a GFF file), then 0 or more characters (.*) until it finds a Note=". Everything up to here is now discarded by the \K, and we only look for the longest stretch of non-" characters ([^"]+).