Linux – /tmp was 1.8Gb; removing files older than 8 days has deleted only ~300mb.. what’s using so much space

I'm running a webserver on CentOS.

Today I couldn't run any SQL queries. Upon investigation, error 28, my /tmp directory couldn't be written to.

I ran du /tmp and it was 1.8Gb. I then ran find /tmp -type f -mtime +8 -exec rm -f {} \; which freed up 300Mb space.

For now, this has done the job – MySQL can operate again.

However, I'm curious as to why the server has so many files in its /tmp directory. I've restarted the server twice and the directory size has remained the same. Don't most linux distributions clear the temp files on reboot?

If I run ls /tmp, I get a massive list of this type of file:

tempCaa7Nw                             tempk7uZU4  temps3GaF1  tempzzWWjV
tempcAbH5o                             tempK7VCdu  temps3hRWv  tempZzXHPM
tempCabZXR                             tempk7VwjK  temps3IdYj  tempzzxZWw
tempcAcQtu                             tempk7W0SI  tempS3IGMF  tempzZy7FM
tempCaCTkx                             tempk7xrzw  tempS3iMfj  tempzZy8L6
tempcAcur7                             tempk7yX3J  tempS3izd5  tempZZYKXr
tempcaCwnz                             tempk81EIE  tempS3JU5s  tempzZyQLO
tempcAdpAk                             tempk81Qpo  tempS3kaNx  tempzZZos7
tempCaeFae                             tempk81ZLV  tempS3KYOx
tempcaEGmt                             tempk83VVy  tempS3LI1u

Full list here.

Do I need to find out what's causing so many temporary files? Or do I simply need to increase the size (or something?) of my /tmp directory? Is that in a config file somewhere?

I'm a poor linux administrator, so please excuse my ignorance.

EDIT for @JeffFerland

Here's the results of grep /tmp on the lsof output file.

mysqld     4649   mysql    4u      REG                7,0         0     103774 /                                          tmp/ib4mpAJ8 (deleted)
mysqld     4649   mysql    5u      REG                7,0         0     103775 /                                          tmp/ibvE7FxN (deleted)
mysqld     4649   mysql    6u      REG                7,0         0     103776 /                                          tmp/ibPe3Lls (deleted)
mysqld     4649   mysql    7u      REG                7,0         0     103777 /                                          tmp/ib656Kb7 (deleted)
mysqld     4649   mysql   11u      REG                7,0         0     103778 /                                          tmp/ibpptQzP (deleted)
cp         7997    root    3r      REG                7,0     18897      28260 /                                          tmp/tempF091j1
cp         7997    root    4w      REG                9,1         0     297690 /                                          tmp_bu/tmp/tempF091j1 (deleted)

ADDITIONAL EDIT
Having deleted ~300Mb of files yesterday, my /tmp directory is back to its original size of 1.8Gb, and once again, MySQL isn't working correctly.

Please help!