The PSP is not UNIX and is not POSIX compliant, however the open source toolchain is composed by gcc 4.3, bintutils 1.16.1 and newlib 1.16.
Most of the C library is already present and can compile most of your code. Lots of libraries have been ported just by invoking the configure script with the following arguments:
LDFLAGS="-L$(psp-config --pspsdk-path)/lib -lc -lpspuser" ./configure --host psp --prefix=$(pwd)/../target/psp
However you might need to patch your configure and configure.ac scripts to know the host mips allegrex (the PSP cpu), to do that you search for a mips*--) line and clone it to the allegrex like:
mips*-*-*)
noconfigdirs="$noconfigdirs target-libgloss"
;;
mipsallegrex*-*-*)
noconfigdirs="$noconfigdirs target-libgloss"
;;
Then you run the make command and hope that newlib has all you need, if it doesn't then you just need to create alternatives to the functions you are missing.