Offset introduction Opamp
Your basic idea is OK.  You have a inverting amp with adjustable gain from 0 to 10.  This gain can be about a adjustable DC level.

However, some of the details need work:

R4 and R3 are silly.  They are each 0.1% of the pot.  That is less than the precision you can set the pot to.  In any case, I can't even guess what you think they do for you.

R2 is worse than silly since it loads the output of R1, and thereby reduces the offset range.  You need the full &plusmn;5 V to get the &plusmn;5 V offset range you specified.

You really should filter the offset voltage.  Stuff happens, and noise can get picked up.  Also, any noise on the +5 and -5 V supplies go directly to the offset voltage, which goes directly to the output signal.

The TL081 has very high input impedance.  You can easily afford to put 100 k&Omega; in series with the output of R1.  That gives you a impedance from 100 k&Omega; to 125 k&Omega; for a cap to work against.  If a human is adjusting the pot, then you can filter down to about 1 Hz.  125 k&Omega; and 1 Hz requires a 1.6 &micro;F cap.  A 1 to 2 &micro;F cap should be fine.  The only drawback of over-filtering is that the result of the pot adjustment will appear to lag.

A TL081 can't put out 10 V with a 10 V supply.  Read the datasheet.  These things need quite a lot of headroom at both ends.  They can take &plusmn;15 V supplies, so all you need is a larger supply range.

Again, you really need to read the datasheet, especially for parts like opamps that have lots of parameters and so many many variations out there.