Hola rodno.
Probá de este modo:
Código:
...
AnsiString name_file = "a.txt";
AnsiString Origen = "C:\\" + name_file;
AnsiString Destino = "E:\\" + name_file;
SHFILEOPSTRUCT FS;
ZeroMemory(&FS, sizeof(SHFILEOPSTRUCT));
FS.hwnd = Handle;
FS.wFunc = FO_MOVE;
FS.pFrom = Origen.c_str();
FS.pTo = Destino.c_str();;
FS.fFlags = FOF_ALLOWUNDO || FOF_NOCONFIRMATION;
SHFileOperation(&FS);
...
Saludos